IN  MEMORIAM 
FLORIAN  CAJORl 


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FIRST  YEAR  IN  ALGEBRA 


BY 

FREDERICK   H.   SOMERVILLE 

THE  WILLIAM   PENN  CHARTER  SCHOOL,   PHILADELPHIA 


NEW  YORK  •:•  CINCINNATI  .:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


Copyright,  1905,  by 
FREDERICK   H.   SOMERVILLE. 

Entered  at  Stationers'  Hall,  London. 


FIRST  year  in  ALG. 

w.  p.  t 


PREFACE 

The  aim  of  this  little  book  is  to  provide  an  introduc- 
tory course  as  a  foundation  to  elementary  algebra.  A 
minimum  number  of  definitions,  an  early  introduction  of 
the  literal  symbol  in  its  simplest  form,  a  clear  conception 
of  the  opposition  of  positive  and  negative  quantity,  and  a 
gradual  introduction  to  the  early  processes  are  believed 
to  be  the  first  essentials  to  successful  later  work.  New 
elements  are  introduced  as  the  result  of  some  natural 
process,  the  exponent,  for  example,  not  being  mentioned 
or  used  until,  in  multiplication,  the  pupil  meets  the  opera- 
tion that  produces  it. 

Certain  important  topics  are  given  a  more  extended 
treatment  than  is  customary  in  most  books  prepared  for 
beginners.  The  application  of  the  equation  to  the  prob- 
lem is  made  in  a  form  that  experience  has  shown  to  give 
excellent  results,  and  the  reasoning  powers  developed  by 
the  limited  classifications  have  been  equal  to  the  demands 
of  the  general  problem.  Substitution  has  a  much  more 
important  position  than  is  usual  in  elementary  teaching, 
and  its  constant  applications  are  designed  to  meet  an 
actual  need  felt  by  teachers  in  higher  grades  of  work. 


4  PREFACE 

The  importance  of  factoring  is  emphasized  by  an  unusual 
amount  of  practice,  and  the  arrangement  of  the  type- 
forms  is 'one  that  has  been  given  extended  trial  in  the 
class  room. 

Reviews  are  constant  and  consist  of  both  classified 
groups  for  topical  review  and  miscellaneous  exercises  in 
which  the  power  of  discrimination  is  first  developed. 
The  frequent  grouping  of  principles  recently  passed  over 
serves  to  keep  them  clearly  in  mind.  The  exercises  have 
been  carefully  graded  throughout  and  are  free  from  diffi- 
cult and  discouraging  questions. 

FEEDEEICK  H.   SOMERVILLE. 

Philadelphia,  Pa, 


■     CONTENTS 

'            OHAPTEE  PAGE 

I.    Symbols 7 

Positive  and  Negative  Quantity      ...  10 

n.    Addition 18 

Parentheses 22 

Review .28 

III.  Subtraction 31 

IV.  Multiplication 36 

V.    Division 49 

Review 58 

VI.    The  Simple  Equation 61 

VII.     Substitution 83 

VIII.     Special  Cases  in  Multiplication  and  Division     .  94 

IX.    Factoring 109 

Review 131 

X.     The  Highest  Common  Factor 140 

XI.    Fractions 143 

Transformations 143 

XII.     The  Lowest  Common  Multiple          ....  151 

The  Lowest  Common  Denominator    .        .        .  154 

XIII.  Fractions  (Continued) 157 

Addition  and  Subtraction  of  Fractions         .  157 

Multiplication  and  Division  of  Fractions     .  161 

Miscellaneous  Fractions 165 

Review 167 

XIV.  Fractional  Equations 174 

XV.    Simultaneous  Simple  Equations        ....  184 

Review 197 


FIRST  YEAR  IN  ALGEBRA 


CHAPTER  I 

SYMBOLS.     POSITIVE   AND  NEGATIVE 
QUANTITY 

1.  In  arithmetic  we  represent  quantity  by  the  use  of 
numbers^  and,  in  most  cases,  we  work  with  units  of  a  par- 
ticular kind. 

Thus,  4  pounds,  3  yards,  and  7  dollars  are  units  of  a  particular 
kind. 

2.  In  algebra  we  represent  quantity  by  the  use  of  the 
letters  of  the  alphabet^  and  we  do  not,  as  a  rule,  assign  a 
particular  value  to  these  literal  symbols. 

Thus,  4  a,  3  m,  6  a:,  and  7  y  represent  what  we  call  general  units, 
and  our  operations  with  them  are  accomplished  without  reference  to 
any  particular  value. 

ILLUSTRATIONS  OF  OPERATIONS  WITH  LITERAL  SYMBOLS 

At  first  the  student  may  wonder  that  operations  with 
these  new  symbols  of  quantity  are  possible,  but  a  com- 
parison of  an  arithmetical  addition  with  particular  units 
and  an  algebraic  addition  with  general  units  will  serve  to 
convince. 

7 


8      SYMBOLS,      POSITIVE  AND  NEGATIVE  QUANTITY 

By  Arithmbtic  By  Algebra 
7  apples  7  a 

5  apples  5  a 

3  apples  3  a 

15  apples  15  a 

No  matter  what  the  '''sl'''  of  the  algebraic  illustration  stands 
for,  it  is  true  that  the  sum  of  seven  a's,  five  a's,  and  three 
«'s  is  fifteen  a's. 

The  principle  is  the  same  when  units  of  dijBferent  kinds 
are  considered  together. 


By  Arithmetic 

By  Algebra 

4  pounds  +  3  ounces 

4a+3& 

6  pounds  +  2  ounces 

6a  +  2& 

10  pounds  +  4  ounces 

lOa  +  46 

20  pounds  +  9  ounces  20  a  +  9  6 

Note  that  in  both  additions,  units  of  the  same  kind  are 
added  together,  pounds  added  to  pounds,  ounces  to  ounces, 
a's  together,  and  5's  together. 

SYMBOLS  OF  OPERATION 

The  principal  signs  of  operation  in  algebra  are  identical 
with  those  of  arithmetic. 

3.  Addition  is  indicated  by  the  "  plus  "  sign,  + . 

Thus,  a  +  b  is  the  indicated  sum  of  the  quantity  a  and  the  quantity 
b.     Read  "  a  plus  J." 

4.  Subtraction  is  indicated  by  the  "  minus  "  sign,  — . 

Thus,  a  —  6  is  the  indicated  difference  between  the  quantity  h  and 
the  quantity  a.    Read  "  a  minus  b." 


SYMBOLS   OF  OPERATION  9 

5.  Multiplication  is  usually  indicated  by  the  absence  of 
sign  between  the  quantities  to  be  multiplied. 

Thus,  ab  is  the  indicated  product  of  the  quantities  a  and  6.  abx  is 
the  indicated  product  of  the  quantities  a,  b,  and  x. 

Sometimes  a  dot  is  used  to  indicate  a  multiplication. 
Thus,  a  •  b  means  the  product  of  the  quantities  a  and  b. 

An  indicated  product  may  be  read  by  the  use  of  the 
word  "times"  or  by  the  use  of  the  literal  symbols 
only. 

Thus,  ab  may  be  read  "  a  times  b."    Or,  simply,  "  ab.** 

6.  Division  is  indicated  either  by  the  sign  of  division, 
-f-,  or  by  writing  in  the  fractional  form. 

Thus,  a  -H  &  is  the  indicated  quotient  of  the  quantity  a  divided  by 
the  quantity  b. 

Also,  -  is  another  form  for  the  same  indicated  operation. 
b 

7.  Equality  of  quantities  in  algebra  is  expressed  by  the 
sign  of  equality,  = ,  which  is  read,  "  equals  "  or  "  is  equal 
to." 

Thus,  to  express  the  fact  that  the  sum  of  two  quantities,  a  and  b, 
is  the  same  as  the  sum  of  two  other  quantities,  c  and  d,  we  write 
a  -{-  b  =  c  +  d. 

8.  A  coefficient  is  a  known  quantity  showing  how  many 
times  another  quantity,  or  group  of  quantities,  is  to  be 
added  in  an  expression. 

Thus,  in  6 a  +  8 xy  "5"  is  the  coefficient  of  «a";  «8"  is  the 
coefficient  of  "  xy." 


10    SYMBOLS.      POSITIVE  AND  NEGATIVE  QUANTITY 

The  student  will  see  that  coefficients  are  results  of  ad- 
ditions, for, 

5  a  is  an  abbreviation  ofa  +  a  +  a  +  a  +  a. 

8  xy  is  an  abbreviation  of  xy  -\-  xi/  +  xy  +  xy  -{-  xy  -{■  xy  +  xy  +  xy. 

The  advantage  of  the  use  of  coefficients  is  evident. 

Sometimes  a  coefficient  is  a  literal  quantity. 

Thus,  in  abx,  "  a  "  may  be  considered  the  coefficient  of  bx,  or 
"  ab  "  may  be  considered  the  coefficient  of  x. 

When  the  coefficient  of  a  quantity  is  "  unity  "  or  "  1," 
it  is  not  usually  written  or  read. 

Thus :  a  is  the  same  as  1  a.     xy  is  the  same  as  1  xy. 

POSITIVE  AND  NEGATIVE  QUANTITY.     SIGNS  OF  QUALITY 

9.  In  algebra  we  consider  not  only  the  amount  of  quan- 
tity, but  its  nature  or  quality  as  well. 

Illustration.  Suppose  a  merchant  has  a  credit  in  a  bank 
amounting  to  1500,  and  suppose,  also,  that  he  owes  the 
sum  of  $500  for  goods  purchased.  Clearly,  the  two  con- 
ditions are  directly  opposite  to  each  other,  for 

the  first  $  500  represents  a  possession, 
the  second  $  500  represents  a  debt. 

Now  possession  is  an  addition  to  the  man's  property ; 
debt,  a  subtraction  from  it :  and  the  fact  that  these  are 
opposites  of  quality  must  be  represented  by  symbols  that 
stand  also  for  direct  opposites.     Hence,  if  we  use 

the  plus  sign  to  indicate  possession, 

we  must,  correspondingly,  use 

the  minus  sign  to  indicate  debt. 


USE  AND   VALUE  OF  QUALITY  SIGNS  11 

So  we  will  write         +  500  representing  his  possession, 
—  500  representing  his  debt. 

From  the  foregoing  we  make  this  important  conclusion : 

10.   As  Signs  of  Quality  the  plus,  +,  and  the  minus,  — , 
symbols  indicate  absolute  opposites  in  kind. 


ILLUSTRATIONS  OP  THE  USE  AND  VALUE  OF  QUALITY  SIGNS 

(a)  If  the  distance  in  an  easterly  direction  from  Chicago 
is  considered  "plus,"  the  distance  west  from  that  city 
would  be  considered  as  "minus." 

west  —  Chicago  4-  east 

(5)  We  may  consider  the  city  of  Memphis,  Tennessee, 
as  in  latitude  about  35°  north  of  the  equator,  or  in  lati- 
tude -f-  35°.  Correspondingly,  Montevideo,  about  35°  in 
latitude  measured  south  from  the  equator,  would  be  con- 
sidered as  in  latitude  —  35°. 

(c)  Suppose  the  temperature  on  a  certain  winter's  morn- 
ing was  reported  as  20°  above  the  zero  point  in  Philadel- 
phia, and  20°  below  the  zero  point  in  Montreal.  We  would 
consider  the  first  as  -}-  20°,  and  therefore  the  latter  as  —20°. 

(d)  If  the  force  expended  to  lift  a  certain  weight  is  100, 
and  the  force  resisting  the  effort  to  lift  it  is  50,  we  would 
express  the  conditions  thus  : 

Force  acting  to  raise  it  +  100. 

Force  acting  against  raising  it  —  50. 

(e)  Suppose  two  men  weigh  exactly  150  pounds.  The 
first  man   gains   10   pounds  while  the   second  loses   10 


12    SYMBOLS.      POSITIVE  AND  NEGATIVE  QUANTITY 

pounds.      The  effects  on  the  men  are  directly  opposite, 
hence  the  quality  of  the  changes  would  be  expressed  thus  : 

First  man's  change  in  weight      +  10  pounds. 
Second  man's  change  in  weight  —  10  pounds. 

Clearly,  therefore, 

11.  Whatever  kind  of  quality  we  indicate  hy  the  plus  sign, 
the  opposite  kind  of  quality  will  be  indicated  by  the  minus 
sign. 

THE  USE  OF  OPPOSITE  QUALITY  SIGNS  TOGETHER 

12.  The  sum  of  a  group  of  plus,  or  positive,  units  of  the 
same  hind  will  be  a  positive  quantity. 

Thus,  if  A  has  three  checks  for  f  100,  $200,  and  $300  respectively, 
we  state  his  possession  : 

+  100 
+  200 
+  300 
+  600  his  possession  in  dollars. 

13.  The  sum  of  a  group  of  minus,  or  negative,  units  of  the 
same  kind  will' be  a  negative  quantity. 

Thus,  if  A  owes  three  different  men  the  sums  of  $100,  $150,  and 
$  200  respectively,  we  state  his  indebtedness : 

-  100 
-150 
-200 

—  450  his  indebtedness  in  dollars. 

Let  us  now  consider  the  effect  of  bringing  together  into 
one  operation  two  or  more  quantities  of  different  quality 
sign ;  that  is,  the  combining  of  positive  and  negative  quan- 
tities in  one  group. 


USE  OF  OPPOSITE  QUALITY  SIGNS  TOGETHER     13 

Suppose  a  man  has  5  dollars  and  owes  2  dollars.     How 
many  dollars  will  he  have  after  paying  his  debt  ? 
By  arithmetic  we  obtain  by  ordinary  subtraction : 

5 
2 
3         • 

But  in  algebra  we  consider  the  quality  of  the  quantities,  hence 
the  possession  is  given  the  +  sign,  consequently,  the  debt  is  given  the 
—  sign.     Then  we  write  ^ 

-2 
+  3 

In  each  case,  clearly,  there  remain  3  dollars.  But  our  algebraic 
process  has  considered  both  amount  and  kind,  and  is  based  on  this 
reasoning : 

His  possession  is +5  or    +1  +  1  +  1  +  1  +  1. 

His  debt  is  —  2   or    —  1  —  1. 

Now  it  requires  one  +  dollar  to  pay  one  —  dollar,  hence  two  +  dol- 
lars are  used  to  pay  the  two  —  dollars.  And  three  +  dollars  remain, 
for  the  algebraic  process  has  considered  kind  as  well  as  amount  in  the 
result. 

Suppose,  again,  that  the  man  has  $  4  but  owes  f  9.  By 
the  same  process :  ,  ^ 

-9 
-5 

For         His  possession  is  +  4  or   +1  +  1  +  1  +  1. 

His  debt  is  -  9   or    -l-l-l-l-l-l-l-l-l. 

Here  five  negative  dollars  remain  after  the  entire  number  of  posi- 
tive dollars  has  been  used  as  far  as  possible  to  pay  dollars  of  debt. 

Again,  the  quality  sign  is  evident  and  of  as  much  importance  as 
the  amount. 


14    SYMBOLS.      POSITIVE  AND  NEGATIVE  QUANTITY 

From  these  illustrations  we  conclude : 

14.  The  algebraic  sum  of  two  quantities  with  different 
quality  signs  is  positive  or  negative  according  as  there  are 
more  +  or  more  —  units  considered. 

With  no  more  difficulty  we  may  group  several  quantities 
of  different  quality  in  one  operation. 

Suppose  that  the  line  AB  represents  the  distance  be- 
tween two  towns,  and  that  (7  is  a  point  midway  between  A 
and  B.  Suppose,  also,  that  distances  to  the  right  of  O  are 
considered  as  -f- .  It  follows,  therefore,  that  distances  to 
the  left  of  Q  will  be  considered  as  — . 

A  O  B 


+  10 

-7    - 

+  12 

> 

A  man  starts  at  (7,  goes  10  miles  towards  B^  turns  back 
7  miles  towards  A^  and  finally  turns  again  and  goes  12  miles 
towards  B.     We  express  his  journey  thus : 

+  10-7  +  12  miles. 

He  traveled  10  miles  and  12  miles  respectively  towards  B;  also 
7  miles  towards  A .     Hence,  his  distance  from  C  will  be 

+  10  +  12  -  7  =  +  15  miles. 

The  result  indicates  not  only  that  he  is  15  miles  from  C  at  the  end 
of  his  journey,  but,  by  its  +  sign,  it  shows  him  to  be  at  the  right 
of  C. 


USE  OF  OPPOSITE  QUALITY  SIGNS   TOGETHER      15 
Suppose  a  different  journey.     Thus : 

A C B 

+  10 


15 


+  8 


-7 


The  several  distances  and  directions  are  expressed  thus : 

+  10-15  +  8-7. 

From  which        +  10  +  8  =  +  18  his  journey  towards  B. 
and  —  15  —  7  =  —  22  his  journey  towards  A. 

Result  —    4  his  final  distance  at  the  left  of  C. 

Using  the  same  line,  AB,  let  the  student  determine  the 
positions  of  the  traveler  at  the  end  of  the  journeys  indi- 
cated in  the  following,  not  forgetting  to  make  use  of  the 
proper  signs  for  locating  him  at  either  the  right  or  the  left 
of  his  starting-point. 

1.  From  C  8  miles  towards  B  and  6  miles  back  towards  A, 

2.  From  C  9  miles  towards  B,  16  miles  towards  A,  and 
back  8  miles  towards  B. 

3.  From  O  6  miles  towards  B,  10  back  towards  A,  and  4 
towards  B. 

4.  From  0 17  miles  towards  A,  11  towards  jB,  6  towards 
A,  and  21  towards  B. 

5.  From  C  7  miles  towards  A,  3  more  towards  A,  15 
back  towards  B^  and  7  towards  A. 


16    SYMBOLS.      POSITIVE  AND  NEGATIVE  QUANTITY 

Exercise  1 

In  each  of  the  following  determine  the  required  numeri- 
cal result,  together  with  its  quality,  whether  positive  or 
negative. 

1.  A  merchant  has  a  credit  of  $1000  in  a  bank  and  gives 
a  check  for  the  payment  of  a  bill  of  |400.  What  is  the 
amount  and  nature  of  his  balance  ? 

2.  The  temperature  on  a  certain  morning  was  58°  above 
zero.  At  noon  it  was  72°  above  zero.  What  quality  change 
had  taken  place  and  how  much  ? 

3.  On  a  winter's  day  at  noon  the  temperature  was  12° 
above  zero.  At  5  o'clock  in  the  afternoon  the  temperature 
had  fallen  19°.  What  was  the  reading  of  the  thermometer 
at  5  o'clock  ? 

4.  A  has  a  cash  balance  of  $45.  He  has  B's  note  for 
115  and  C's  check  for  1 30,  and  he  owes  D  |100.  How 
will  A  stand  when  he  balances  his  account  ? 

5.  A  has  $100.  B  has  no  cash  whatever.  C  also  has 
no  cash,  and  yet  owes  8 100.  How  much  better  off  than 
B  is  A  ?  How  much  better  off  than  C  is  B  ?  How  much 
better  off  than  C  is  A  ? 

ALGEBRAIC  EXPRESSIONS 

15.  An  Algebraic  Expression  is  an  algebraic  symbol,  or 
group  of  algebraic  symbols,  representing  some  quantity. 

An  expression  is  called  numerical  when  made  up  wholly 
of  numerical  symbols. 

Thus,  20  +  10  —  13  is  a  numerical  expression. 


ALGEBRAIC  EXPRESSIONS  17 

An  expression  is  called  literal  when  made  up  wholly  or 
in  part  of  literal  symbols. 

Thus,  ab  +  mn  —  xy  is  a  literal  expression. 
5a  +  3c  —  17  is  a  literal  expression. 

16.  The  Terms  of  an  algebraic  expression  are  its  parts 
connected  by  the  +  or  —  signs. 

Thus,  in  the  expression  ab  +  mn  —  xy, 

ab,  +7nn  and  —  xy  are  terms. 
It  will  be  noted  that  the  sign  between  two  terms  belongs  to  the  term 
following  it.     Also  that  the  sign  of  the  first  term,  when  +,  is  not 
usually  written. 

Thus,  in  the  expression  ab-\-mn  —  xy, 
the  first  term  is  +  ab. 

Y7.   The  common  expressions  of  algebra  are  named  as 
follows  : 

Monomial.     One  term. 

3  a,  10  mn,  27  xyz  are  monomials. 
Binomial.      Two  terms. 

a  +  b,^m  —  x,10  —  lcd  are  binomials. 

Trinomial.     Three  terms. 

3w  +  7n  —  Sxyisa  trinomial. 

Polynomial.     A  name  usually  given  to  expressions  of 
more  than  three  terms. 


p.  H.  S.  FIRST  YEAR  ALG.  —2 


CHAPTER   II 
ADDITION.     PARENTHESES.     REVIEW 

18.  Addition  in  algebra,  as  in  arithmetic,  is  the  process 
of  combining  two  or  more  expressions  into  an  equivalent 
expression  called  the  sum. 

In  algebraic  addition  we  consider : 

(1)  Like  quantities  having  the  same  signs,  either  all  + 
or  all  — . 

(2)  Like  quantities  having  different  signs,  some  +  and 
some  — . 

When  the  quantities  to  be  added  are  like  in  sign,  the 
addition  is  in  no  way  different  from  that  of  arithmetic. 
Thus : 

By  Arithmetic  By  Algebra 

7  books  +    7  a                -7a 

10  books  +  10  a                 -  10  a 

15  books  +  log                 —15a 

32  books  +  32  a                -  32  a 

From  which  we  conclude  : 

19.  77ie  coefficient  of  the  sum  of  similar  terms  having  like 
signs  is  the  sum  of  the  coefficients  of  the  terms  with  the  com- 
mon sign. 

When  the  quantities  to  be  added  are  unlike  in  sign,  the 
resulting  coefficients  depend  for  their  sign  upon  the  excess 
of  one  quality  sign  over  the  other.     (Art.  14.) 

18 


GENERAL  METHOD  19 

Applying  this  principle  to  the  coefficients  of  like  terms, 
we  will  obtain  the  sum  of  the  following : 

5a4-  7&  +  2c 
2a-3&-5c 


7a  +  46-3c    Result. 

The  result  is  obtained  as  follows : 

fThe  coefficient  "  +  7 "  in  the  1st  term  is  the  sum  of  the  given 
coefficients,  +  5  and  +  2. 

The  coefficient  "  +  4  "  in  the  2d  term  is  the  sum  of  the  given  coeffi- 
cients, +  7  and  —  3. 

By  Art.  14  this  coefficient  is  the  actual  difference  between  7  and  3, 
with  the  sign  of  the  greater. 

The  coefficient  "  —  3  "  in  the  3d  term  is  the  sum  of  the  given  coeffi- 
cients, +  2  and  —  5. 

Here,  as  above,  the  coefficient  "—3"  is  an  actual  difference  with 
the  proper  sign. 

From  this  illustration  we  conclude  : 

20.  The  coefficient  of  the  sum  of  similar  terms  having  un- 
like signs  is  the  actual  difference  between  the  sum  of  the  4- 
coefficients  and  the  sum  of  the  —  coefficients^  with  the  sign 
of  the  greater. 

GENERAL  METHOD 

The  following  method  is  in  general  use  for  the  addition 
of  expressions  made  up  of  similar  terms. 

Ex.  1.    Add  ^a  +  lh-2c,  2a-35  +  8(?,  and  -3a-t- 

5a  +  7&  -    2c 

2a-3&  +    8c 

~3a  +  2ft-10c 

4a  +  66-   4c    Result. 


20  ADDITION.      PABENTHESES.      REVIEW 

The  coefficients  of  the  result  are  obtained  as  follows : 

Sum  of  the  +  coefficients  of  a,  +  7 ;  sum  of  the  —  coefficients  of 
a,  —  3.     Difference,  +  4. 

Sum  of  the  +  coefficients  of  &,  +  9 ;  sum  of  the  —  coefficients  of 
ft,  -  3.     Difference,  +  6. 

Sum  of  the  +  coefficients  of  c,  +  8;  sum  of  the  —  coefficients  of 
c,  —  12.     Difference,  —  4. 

Remember  that  the  signs  of  the  first  terms,  5  a  and  2  a,  are  "  plus 
understood."  • 

If  not  all  of  the  terms  considered  are  represented  in  each 
given  expression,  it  is  well'  to  arrange  in  alphabetic  order, 
leaving  space  for  such  terms  as  an  examination  of  the 
problem  shows  need  for. 

Ex.  2.  Add4a  +  35  +  3w,  26  +  3(?-6?,  2a+3(^+2w-a;, 
and  5b  —  5m—Sx, 

4a  +    3&  +  3m 

+    2b  +  Sc-    d 
2a  -\-Sd  +  2m-    X 

+    5&  — 5m  —  3a? 

6a  +  10b  +  Sc  +  2d  -4a:  Result. 

Note  that  the  sum  of  the  coefficients  of  "  m  "  is  zero,  and,  there- 
fore, the  m  term  disappears  in  the  result. 

We  are  now  ready  to  state  the  general  process  for  addi- 
tion of  algebraic  expressions. 

21.  Place  like  terms  in  vertical  columns. 

Add  separately  in  each  column  the  positive  coefficients  and 
the  negative  coefficients^  and  to  the  arithmetical  difference  of 
their  sums  give  the  proper  sign. 


GENERAL  METHOD  21 

22.  Collecting  terms  in  algebra  is  another  expression 
for  addition.  It  is  frequently  used  in  cases  where  like 
terms  are  scattered  throughout  an  expression  and  in  no 
particular  order. 

Exercise  2 

Find  the  sum  of : 

1.  2.  3.  4.  5.  6.  7.  8. 

a  -a  10a  -10a  -Sa  -Ua  87a;  -29y 
la     -la     -6a  6a       15a     -14a     -18a;        13^ 

9.  10.  11.  12.  13. 

2a  +  4h     Za-\-lx     —Im—lxz     —8a;— 13     —aed-\-   ^yz 
ba-2h        a-6x         4m-\-^xz       lla;+   1      ^acd-llyz 

14.  15.  16. 

bxy-\-'^xz—    yz  —   2nx—10     p  —ah  —x-\-l 

2>xy  4-2?/2       Smy-\-llnx  bab—2cd-\-x 

17.  3a+26,  4a-35,  -2a+75,  and  5a-6h, 

18.  2a— 3<?+a;,  3a  +  4(?  +  3a;,  and  a  —  3 c  —  2 a;. 

19.  5a;+3y-10,   -6a;  +  2«/-9,  and  x-2y^l, 

20.  4m  —  37^^-2a?,   —2n  —  x^  Sm  —  n,  and  3n—Sx. 

21.  2a-35— 4c— 5,  a  +  7h+^c—2,  and  -7a+35-c. 

22.  — 3a  +  35-3c  +  6,  c  — 25  +  5  — a,  and 
—  5  —  254-5flj  —  <?. 

23.  a-2y  +  h-Sx+10,  2h -S  y -\-2  a-7, 
3  a;  —  2  a  +  y,  and  45— 3a;— 2a—  11. 


22  ADDITION.      PARENTHESES.      REVIEW 

Collect  similar  terms  in  each  of  the  following : 

24.  6  ah -\- She  — 2  mn  —Zx-\-2hc  —  cd  —  ah-\-%  mn 

—  ah  —  S  cd  -{-  X. 

25.  3w+7  — a  — a;  +  8  — m+lla;  — 15a  — 4w 

—  2n-^Sm  —  Sx. 

26.  2ahc—5  hed -i- 2  cdx -11-^  15  hcd  —2abc 

—  cdx  —  2  +  ahc. 

27.  —2a  +  Qac-\-ab—^ac—a—bah-\-^a—2ac-\-2>ah. 

28.  Zd-[-a-d-\-c-'^d-2c+4:h-\-c-^h  +  lQa-2a. 

29.  — a  —  m  +  c— x-\-c  —  m-\-a  —  y  —  h-{-m  —  c  —  a 
+  m  +  h-\-y  +  a  —  c  +  h-{-x, 

THE  PARENTHESIS 

23.  The  Parenthesis  is  used  in  algebra  to  indicate  that 
two  or  more  quantities  are  to  be  treated  as  a  single  quan- 
tity.    Thus : 

a  +  (ft  —  c)  means  that  to  a  we  are  to  add  the  difference  of  h  and  c. 
a  —  (b  —  c)  means  that  from  a  we  are  to  subtract  the  difference  of 
b  and  c. 

24.  A  sign  hefore  a  parenthesis  indicates  an  operation. 
A  +  sign  hefore  a  parenthesis  is  an  indicated  addition. 
A  —  sign  hefore  a  parenthesis  is  an  indicated  suhtraction. 

We  will  first  consider 

THE  PARENTHESIS  PRECEDED  BT  THE  PLUS  SIGN 

Suppose  we  add  to  20  the  sum  of  10  and  5. 
We  write  20 +(10 +  5). 


PARENTHESIS  PRECEDED  BY  MINUS  SIGN         23 

From  this  indicated  addition  we  obtain 

20  +  (10  +  5)=  20  +  15         20  +(10  +  5)=  20  +  10  +  5 
=  35  ^^'  =  35 

The  result  is  the  same  whether  the  10  and  5  are  added  before  or 
after  removing  the  (  ). 

Suppose,  again,  that  we  add  to  20  the  difference  between 
10  and  5. 

We  write  20 +(10 -5). 

And,  as  before, 

20  +(10  -  5)=  20  +  5        20  +  (10  -  5)=  20  +  10-5 
=  25  "^'  =25 

The  same  result  in  either  case.     Hence,  we  conclude : 

25.  if  a  +  (  )  ^s  removed^  none  of  the  signs  of  its  terms  is 
changed. 

"The  parenthesis  preceded  by  the  minus  sign 
Suppose  we  subtract  from  20  the  sum  of  10  and  5. 
We  write  and  obtain 

20  -  (10  +  5)  =  20  -  15         20  -  (10  +  5)  =  20  -  10  -  5 
=    5  ^^'  =5 

The  same  result  in  either  case. 

Suppose,  again,  that  we  subtract  from  20  the  difference 
between  10  and  5. 

We  write 

20  -(10  -  5)=  20  -  5         20  -(10  -  5)  =  20  -  10  +  5 
^15  "''  =15 

Again,  the  same  result  in  either  case.  Clearly,  in  the  last  operation 
we  are  not  to  take  all  of  10  from  20 ;  only  the  difference  between  10 
and  5  is  to  be  taken.  So  if  10  is  taken  away,  5  must  be  added,  and 
we  have  20  -  10  +  5,  or  15. 


24  ADDITION.      PARENTHESES.      BEVIEW 

We  may  conclude,  therefore,  that 

26.  If  a  —  (^^  is  removed^  the  sign  of  every  term  in  it  must 
he  changed. 

It  must  be  remembered  that 

{a)  The  sign  before  a  parenthesis  indicates  an  operation 
of  either  addition  or  subtraction. 

(5)  The  sign  of  a  parenthesis  disappears  when  the  paren- 
thesis is  removed. 

The  effect  of  all  possible  cases  of  +  and  —  signs  before 
parentheses  is  clearly  illustrated  as  follows : 

+  (+«)  =  +  «  —(+«)=—« 

PARENTHESES  WITHIN  PARENTHESES 

It  is  frequently  necessary  that  we  inclose  in  parentheses 
parts  of  an  expression  already  in  another  parenthesis.  In 
order  to  avoid  confusion  in  such  cases,  we  employ  different 
forms  of  the  parenthesis.     The  forms  are : 

(a)  The  Bracket  [  ].  (6)  The  Brace  \\.  (js)  The 
Vinculum  "  .  Each  has  the  same  significance  as  the 
ordinary  parenthesis.     Thus: 

(a  +  6)  =  [a  +  5]  =  [a  +  ^1  =  a-\-h. 

In  removing  one  or  more  of  these  signs  of  aggregation, 
we  observe  the  following  : 

27.  Beginning  with  the  innermost.,  remove  the  parentheses 
one  at  a  time. 

If  a  parenthesis  preceded  by  a  minus  sign  is  removed,  the 
signs  of  its  terms  are  changed* 


PARENTHESES    WITHIN  PARENTHESES  25 

After  removing  all  parentheses,  collect  the  terms  of  the 

result. 

Exercise  3 

Observing  the   effect  of  the  +  and  —  signs  preceding 
parentheses  that  you  remove,  simplify  the  following : 
1-    (+5)+  (+2)  =  5  +  2 

.  =  7.     Result. 

2.  (+5) -(+2)  =  5- 2 

=  3.     Result. 

3.  (+5) +  (-2).  8.  (+16) +  (-17). 

4.  (+5) -(-2).  9.  (-14) -(-13). 

5.  (-7)  +  (  +  6).  10.  (-7a)-(+5a). 

6.  (+5)-(-9).  11.  _(-5a)-(+7a). 

7.  (_10)-(_19).  12.  -(+6w)  +  (-6m). 

13.  (-13:r?/)  +  (-llrry). 

14.  —  (— 19mn)  — (+20  wn). 

15.  3m-(2m)  +  (-m)  +  (2m)  =  3m-2w-wj  +  2m 

=  2m.    Result. 

16.  (+6) -(+3) +(-2). 

17.  (-5) -(-2) -(-3). 

18.  a— (4a) +  (—«)  — (—4a). 

19.  -5x+(-2x')-(-x}-{Sx'). 

20.  a  -  [2  6  +  (3  c  -  2  6)  -  c]  =  a  -  [2  6  +  3  c  -  2  6  -  c] 

=  a-2b  -Sc  +  2b  +  c 
'  =  a  —  2  c.    Result. 

21.  5a  +  (3a  +  ll).  26.    -  3a;-(-rr  +  «/-9). 

22.  4a-(3a  +  6).  27.    4:r+  [22:  +  (2:  +  l)]. 

23.  3a;+(2a;-3)+4.  28.   4a^- [2  2;-(a:- 1)]. 

24.  5:r-(3a;-^  +  l).  29.    3  ^/ -  jy  +  (2  y-10)  j. 

25.  6c-(2c?  +  3)-10.  30.    2a;?/-  [a;2/  +  3-2;?/]. 


26  ADDITION,      PARENTHESES.      REVIEW 


31.  by+i\l-\2y-y  +  l\-], 

32.  -7-(-w+ [-3m- J2m  +  ^n)- 
33. 


4  ?n  -  [3  m  -  (w  +  2)  -  4]  -  {m  +  3  n  -  (n  -  1)}  +  n 


4m- [3 w-  n-2  -4]-{m  +  3n-  n  +  1}  +n  = 
4m-  3m+  n  +  2  +4  -{m  +  3n-  n  +'l}  +  n  = 
4m—    3m+    n  +  2    +4    -m-3n+    n-1     +w  =  5.     Result. 


34.    a  —  [a  +  f a— (<x  — a  — 1)1] 


35.  — 3m+J  —  m  +  (— 2m  —  6  —  w)J. 

36.  5tf-[3  +  (-2c-  [(?_(3(?+ J(?-70])] 


37.    (4  a  -  1)  -  [2  a  -  (a  +  1)  -  3  a  -  7]  -  (8  -  a  +  1) . 


38.    l_(_l)-j_l4-l_a4-(-l)|. 


39.    l+(-l)  +  ;-l-l_a-(-l){. 


40.    a-\-l—a  +  \-a-\-Q-a^a-V)\']. 


41.    a  — [— a  — J  — a  — (— a  — a  — l)j]. 


42.   «— 5(5  — c  — a-c  +  l)— a  — ft  — 1)J. 


43.  a-  [a-a-(a  +  l)-  [a-(l-l-a  +  2a)]5. 

44.  l_(_l)-5-(_l)j_j_(_;_(_i)_l|)|_l. 

INCLOSING  TERMS  IN  PARENTHESES 

It  is  often  necessary  to  group  certain  terms  of  an  alge- 
braic expression  so  as  to  treat  the  group  as  one  term. 
This  operation  is  the  exact  opposite  of  the  removal  of 
parentheses,  hence  we  invert  the  statements  of  Arts.  25 
and  26,  and  obtain  the  following  : 


INCLOSING   TERMS  IN  PARENTHESES  27 

28.  An^  number  of  terms  may  he  inclosed  in  a  -\-  (^^  with- 
out ehanging  the  signs  of  the  terms  inclosed. 

29.  Any  number  of  terms  may  be  inclosed  in  a  —  (^)  pro- 
vided the  sign  of  each  term  inclosed  is  changed. 

Exercise  4 

Insert,  in  each  of  the  following,  a  parenthesis  inclosing 
the  last  two  terms,  each  parenthesis  to  be  preceded  by 
the  plus  sign. 

1.  a  +  2  6  +  7c.  4.   l%ab  +  ^bc-2cd-M. 

2.  5  a  +  3  w  + 2  w— 7.  5.   Zm  —  -\:n  —  bp  —  xyz. 

3.  ^xy-\-l  xz-\-^yz—^,         6.    -^  a  +  lbb-10-\-yz. 

Insert,  in  each  of  the  following,  a  parenthesis  inclosing 
the  last  three  terms,  each  parenthesis  to  be  preceded  by  a 
minus  sign. 

7.  3a- 5n  +  7;)-19  =  3a- (5n-7j9  + 19).     Result. 

8.  a6c  — 4  cc?  +  wn  — njo  — 3  =  a6c  — 4  cc?— (  — mn  +  n/?  +  3).     Result. 

9.  x  —  y-{-z  —  \.  12.    — 10  — 7;?  +  8n  — 5w. 

10.  3^- 2  5-4  c  +  ^.  13.   4:a-\-b-\-c-^d. 

11.  ^ab-1bc  +  Qcd-^0.         14.    16-W  +  3  w-3jt?  +  a;2. 

15.  a  +  b-c-\-d-e+f 

16.  —  3  —  w  +  mn  ~np  —  xyz, 

17.  x—b  a  —  y  ■\'Z  —  1, 

18.  5aH-2^— 3a;— 5. 

19.  ac  —  be  —  ah  +  ax  —  ay. 


28  ADDITION.      PARENTHESES.      REVIEW 

GENERAL  REVIEW  — GROUP  I 

SIGNS  OF  OPERATION 

The  +  sign.     An  addition  or  increase. 
The  —  sign.     A  subtraction  or  decrease. 


SIGNS  OF  QUALITY 

sign. 


^,  .       r  Absolute  opposites  in  kind. 

The  —  siqn.  J 


ADDITION 

The  sum  of  a  group  of  all  +  units  of  the  same  kind  is  a 
+  quantity. 

The  sum  of  a  group  of  all  —  units  of  the  same  kind  is  a 
—  quantity. 

The  sum  of  a  group  of  both  +  and  —  units  of  the  same 
kind  is  either  -\-  or  —  in  sign  according  as  there  are  more 
+  or  more  —  units  considered. 

PARENTHESES 

The  signs  before  paren- 1.        .  ^  .  ._.        _,,.. 

^,  .  A  A  -\-  (  )  IS  an  indicated  addition, 

theses    are    signs    of  \    .       ;r.         ._.        __ 

\  A  —  (  )  IS  an  indicated  subtraction, 
operation.  J  ^ 

(a)  Removal 

If  a-\-(^)is  removed.,  none  of  the  signs  of  its  terms  is  changed. 
If  a— (^}  is  removed^  all  of  the  signs  of  its  terms  are  changed. 
The  sign  before  aparenthesis  disappears  with  the  parenthesis. 

(b)  Insertion 

Inclose  an  expression  m  a  4-  (  )  and  change  no  signs  of 
terms  inclosed. 

Inclose  an  expression  in  a  —  (^^  and  change  the  sign  of 
each  term  inclosed. 


GENERAL  REVIEW  29 

Exercise  5 

1.  A  merchant  has  1500  and  f  700  in  two  banks  and 
owes  $  800  for  goods  purchased.  What  is  the  expression 
for  his  actual  financial  condition  ? 

2.  If  the  same  merchant  had  a  dollars  in  the  first  bank, 
b  dollars  in  the  second  bank,  and  owed  c  dollars,  what  would 
be  the  expression  of  his  possession  ? 

3.  On  a  certain  day  the  temperature  rose  20°  from  the 
zero  point,  but  at  sunset  it  fell  13°.  Express  the  change 
and  the  result  with  proper  signs. 

4.  If  a  thermometer  indicates  a  rise  of  a°  and,  later,  a 
fall  of  x°,  how  shall  we  express  the  final  result  ? 

5.  A  man  weighs  150  pounds,  but  loses  m  pounds. 
Later  he  gains  n  pounds.     What  is  his  final  weight  ? 

'*    6.   Express  with  proper  sign  the  result  of 

(+3)-(+2)-(-3)  +  (-5)  +  (+2). 

7.  What  is  the  sum  of 

_(_2)_(-3)  +  (-2)-(+3)? 

8.  Is  the  sum  of 

[_3  +  (_2)-(-l)]-[_(-2)  +  C-l)] 
positive  or  negative  ? 

9.  Prove  that 

[3-(-2)-(l)]  +  [-2-(-3)  +  (-5)]=0. 

10.  A  boy  claims  that 

[(-  2)  -  (+  3)  -  (-  5)]  is  the  same  as 
[-(-2) -(-3) -(5)].     Is  he  right? 

11.  Find  the  sum  of5a  +  3&  —  m,  25  —  3<?  +  a;,  — 4a  — 
5 6  +  5 c  +  Wj  and  a  —  2c  —  x. 


30  ADDITION.      PARENTHESES,      REVIEW 

12.  Collect  -2a  +  3c-4(^,  4(?-a;+7,  4a— 6c-h3(?— 4, 

and  —a-\-2d  +  1x  —  ^, 

13.  Simplify  and  collect 

a  +  [  -  2  (7  -  3  a  +  (5  -  c  +  a)  +  5] . 

14.  Simplify  and  collect 

1-!1  +  (1-[1  +  1]-1)}. 

15.  Simplify  and  collect 

10  _  [9  +  8  -  (7  +  6  -  55 -f  4  -  3T2S  - 1)]. 

16.  In  a  ball  game  the  Bostons  made  —  [2  —  (3  +  2)] 
runs,  and  the  New  Yorks  made  —3  — [—(2  +  3)]  runs. 
How  many  runs  were  made  in  all  ? 

17.  A  boy  was  asked  to  write  his  age  on  a  slip  of  paper, 
and  he  wrote  "  —  (+[—  12 +  (—  3)] )  years  of  age."  How 
old  was  he  ? 

18.  Simplify  and  collect 

-  [2  -  ^"^]  -  S  -  (-a  -  [- a  -  r=^]){. 

19.  Inclose  the  last  four  terms  of  the  following  in  a 
parenthesis  preceded  by  a  minus  sign  : 

a  —  3J4-4m  —  5/1  +  27. 

20.  Inclose  the  last  three  terms  of  the  following  in  a 
parenthesis  preceded  by  a  plus  sign : 

ax  —  am  —  mx  +  mt/  —  ay, 

21.  Beginning  at  the  left,  inclose  each  pair  of  terms  in  a 
parenthesis,  the  first  two  parentheses  to  be  preceded  by 
plus  signs  and  the  last  two  preceded  by  minus  signs. 

<i  —  Z'b  —  ^c-\-d  —  m  —  n-\-x  —  y, 


CHAPTER  III 
SUBTRACTION 

30.  Subtraction  is  the  process  of  taking  one  quantity 
from  another  quantity. 

The  quantity  taken  away  is  called  the  Subtrahend. 

The  quantity  from  which  the  subtrahend  is  taken  is 
called  the  Minuend. 

The  result  obtained  by  taking  the  subtrahend  from  the 
minuend  is  called  the  Difference. 

31.  The  general  process  of  subtraction  depends  upon 
the  principles  already  explained  in  Art.  26.     For, 

The  quantity  taken  away  may  he  considered  as  inclosed  in 
a  parenthesis  preceded  hy  a  minus  sign. 

Thus,  suppose  we  write  : 

From  10a  +  36  +  7c  take  6  a  +  5  +  <?. 

The  first  expression  is  our  minuend  and  the  second  expression 
our  subtrahend. 

Consider  each  expression  a  single  quantity  and  inclose  each  in  a 
parenthesis. 

Then,  using  the  minus  sign  of  operation  for  the  word  "  take,"  we 

^^^®  (10  a  +  3  &  +  7  c)  -(6  a  +  &  +  c). 

Removing  both  parentheses, 

10a  +  3&+7c-6a-&-c. 

Collecting,  4a  +  2&  +  6c.     Result,  or  Difference. 

31 


32  SUBTRACTION 

We  have,  therefore,  changed  the  signs  of  the  terms  of  the  original  sub- 
trahend and  collected  all  terms.     The  same  operation  might  have  been 

written  thus:  m^  .  qt.  •  '7       -Mf-         j 

10a  +  «3o+7c    Minuend. 

6  a  +     ft  +    c    Subtrahend. 


4a  +  26  +  6c    Diif  erence. 

The  student  must  bear  in  mind  that  the  signs  of  the 
subtrahend  are  not  actually  changed.  They  are  only  con- 
ceived to  he  changed.  At  first  the  student  will  be  greatly 
assisted  by  indicating  the  conceived  changes  in  his  work, 
but  he  should  immediately  form  the  habit  of  making  the 
change  mentally.  The  given  signs  of  the  subtrahend  should 
never  be  altered  in  any  way. 

From  the  foregoing  we  will  now  state  the  general  pro- 
cess for  subtraction. 

32.   To  subtract  algebraic  expressions : 

Place  similar  terms  in  vertical  columns. 
Consider  the  sign  of  each  term  of  the  subtrahend  to  he 
changed^  and  proceed  as  in  addition. 

Before  taking  up  the  practice  of  algebraic  subtraction 
the  student  should  carefully  note  the  following : 

(a)  Subtracting  a  positive  quantity  is  the  same  as  adding 
a  negative  quantity. 

For,  (+5)-(+3)  =  +5-3  =  2, 

and  (  +  5)  +  (  -  3)  =  +  5  -  3  =  2. 

(6)  Subtracting  a  negative  quantity  is  the  same  as  adding 
a  positive  quantity. 

For,  (+5)-(-3)=  +  5  +  3  =  8, 

and  (+  5)  +  (+  3)=  +  5  +  3  =  8. 


SUBTRACTION  33 

These  results  follow  directly  from  the  simple  removal 
of  parentheses. 

Exercise  6 


1. 

2. 

3. 

4. 

5. 

6. 

From 

+  10 

-\-12a 

-6a 

-3a 

+   Sah 

+  5a; 

Take 

4-  4 

+  5a 

-2a 

-2a 

-lab 

-Sx 

+   6 

+   7a 

-\a 

—    a 

+  10  a6 

+  Sx 

7. 

8. 

9. 

10. 

11- 

12. 

From 

10  m 

-    le 

6xy 

-%ac 

— 4mw 

—  5a:0 

Take 

12m 
-2m 

4:0 
-lie? 

-\xy 
V^xy 

-2ae 

—  9mn 

19  xz 

—    a<? 

-24:XZ 

13. 

14. 

15. 

From 

4 

a-\-6 

7 

m-13 

— 

13a:-7 

Take 

2 

a-3 

5 

m-18 

- 

-6^  +  4 

16.                         17.  18.  19. 

Sa  +  ^h  +  lO  lm-4:n  -2b           +6  0 

25  +  7  3m-2n  +  7  2a-25  +  3m  +  6  a  +  b-{-c 

5a+    5  +  3  4m  — 2n— 7  —2a          —3m  —a—b  —  e 

20.  From  5  a  +  4  m  —  7  take  3  a  —  m  +  3. 

21.  From  2  a  +  4  a;  —  ^  take  a  —  5  a;  —  3  y. 

22.  From  3  a;  +  7  ?/  —  10  take  —x—y  +  6, 

23.  From  4iX  —  By—Zz  take  2  «/  +  8  3. 

24.  From   -3a  +  4c-2(Z4-7  take  3a+7<7-2. 

25.  From  4  a  +  11  m  —  a:  take  3  a  +  7  a:  —  ?/. 

26.  From  35  +  2c  —  4m  take  — a— 5  +  2c  —  m. 

F.  H.  S.  FIRST  YEAR  ALG.  —  3 


34  SUBTRACTION 

27.  Subtract  12  ah -2  be -10  from  11  ah  -  4  ho -{- S. 

28.  Subtract  10-\-Sx-2i/  +  z  from  15  — 4:x-22/  +  z. 

29.  Subtract  5m-^S  n  —  2  y  -{-z  from  2m—n+22/  —  z. 

30.  Subtract  6a  +  4J-3(?  from  6a-46+3c-12i. 

31.  From  the  sum  of  2m-\-2n  —  p  and  3w  —  Sri  +  Sjt? 
subtract  the  sum  of  m-^n—2p  and  Sm—  2n  —p. 

2m  +  2n—     j9  wi+     n  —  2^?        5  m  — n  +  2j9 

3  m  — 3n+3jo         3  m  — 2n—     p        4:7n  —  n—Sp 
5m—     n  +  2 ji?        4:771  —     n  —  3p  m         +  5jt>  Result. 

32.  From  the  sum  of  2  a  +  ih  —  c  and  3a  —  45  + 3  c 
take  the  sum  of  S  a  +  5h—l  c  and  2a  —  4:h-\-5c. 

33.  From  the  sum  of  3a— 55  +  10  and  ih  +  l  c—7 
take  the  sum  of  6  +  8  c  —  12  a  and  16  5  —  19  +  6  a  —  4  (?. 

34.  Take  the  sum  of  4w  +  37i  —  5?/,  Sn  +  4:i/  —  z,  and 
w+^  — 14  2  from  the  sum  of  2m -{-Si/,  ^n  —  x,  5p-\-z, 
and  —  12  w  —  9. 

ADDITION  Ain)  SUBTRACTION  WITH  DISSIMILAR  COEFFICIENTS  * 

33.  Thus  far  our  additions  and  subtractions  have  been 
made  when  the  coefficients  of  like  terms  were  all  numeri- 
cal and  easily  combined.     Thus : 

5x  15  ar 

3a:  6a; 

8  a;  Sum.  9  a;  DifEerence. 

In  later  work  it  will  often  happen  that  coefficients  will 
be  literal  and  not  possible  of  combination  into  a  simple 

♦  At  the  teacher's  discretion  this  subject  may  be  omitted  until  Exer- 
cise 47  is  reached. 


ADDITION  AND  SUBTRACTION  35 

form  like  a  numerical  sum.     In  such  cases  our  addition  is 
indicated  and  the  process  is  as  follows : 


ax 

bx 

mx 
nx 

2  ax 
^cx 

5  am 
—  7  cm 

(a  +  b)x 

(m  -\-  n)x 

(2a- 

f  3c)a; 

(5  a  -  7  c)  m 

Add  the 
1. 

Exercise  7 
following : 
2. 

3. 

4. 

ay 

cy 

Sx 

mx 

( 

xyz 

—  axyz 
)xyz 

cm 
—  ahem 

(         )^ 

(     > 

(         ^cm 

ax 
nx 

5. 

+      So       + 
-       he       - 

fan 
an 

6. 

5771+    w  4-        X 
-am-\-en—S  ax 

(a  +  n)  a;  -h  (3  —  5)  <?  +  (w  —  a)  w  (         )  wi  +  etc. 

7.  ah  +  mn,  4  5  —  en,  and  h  -{-  n. 

8.  5  m  -{-  Sn  -{-  ax  and  2  am  —  4:  en  —  x. 

9.  4  6c  +  5  5c^  +  3  /wTi,  2  (?  —  hmn,  and  —  ac  —  ahd. 


Subtract  the  following : 

10.                                 11. 

12. 

mx                             hay 

ma       + 

a; 

nx                          —Zey 

3a       - 

ax 

{m  —  n)x  (ba-\-^c)y  (         )«+(         )^ 

13.  Subtract  4  a;  —  3  ^  from  ax  +  %. 

14.  Subtract  5  m^i  —  7  aa;  from  am/i  +  hx. 

15.  Subtract  3  my  -{-2nx—Sz  from  a?/  +  aa;  +  a;?. 


CHAPTER  IV 

MULTIPLICATION 

34.   Multiplication  is  the  process  of  taking  one  quantity 
as  many  times  as  there  are  units  in  another  quantity. 
Multiplication  is  an  abbreviated  form  of  addition,  for 
3x4  =  4  +  44  4.  5a  =  a  +  a  +  a  +  a-{-a. 

The  Multiplicand  is  the  quantity  to  be  multiplied. 

The  Multiplier  is  the  quantity  showing  how  many  times 
the  multiplicand  is  to  be  taken. 

The  multiplier  shows  also,  by  its  sign,  how  the  multi- 
plicand is  to  be  taken;  that  is,  whether  to  be  added  or 
subtracted.     Thus, 

5  multiplied  by  +  3  indicates  a  positive  or  additive  result,  and  5 
multiplied  by  —  3  indicates  a  negative  or  subtractive  result. 

The  Product  is  the  result  of  a  multiplication. 

SIGNS  m  IMULTIPLICATION 

It  has  been  stated  that  the  multiplier  shows  not  only 
how  many  times  a  quantity  is  to  be  taken,  but  how  it  is  to 
be  taken  as  well. 

The  following  will  illustrate : 

(I)  Positive  Multiplier.     A  product  to  be  added. 

8  times  5  expressed  with  all  signs  gives 

+  3x  +  5=+5  +  5  +  5  =  +  15. 
3  times  —  5  expressed  with  all  signs  gives 

+  3x-5  =  -5-5-5  =  -15. 


EXPONENTS  IN  MULTIPLICATION  37 

(II)  Negative  Multiplier.     A  product  to  be  subtracted. 

—  3  times  5  expressed  with  all  signs  gives 

_  3x  +  5  =  -  (  +  5+5+5)  =  _  (+  15)  =  -  15. 

-  3  times  —  5  expressed  with  all  signs  gives 

_  3  X  -  5  =  -  (-  5  -  5  -  5)  =  -  (-  15)  =  +  15. 

By  comparing  these  four  possible  cases  we  have 

+    5  -5  +5  -5 

+    3  +    3  -    3  -    3 

+  15  -  15  - 15  +15 

From  which  we  make  these  important  conclusions : 

35.  Like  signs  in  multiplication  give  -f . 

36.  Unlike  signs  in  multiplication  give  — . 

COEFFICIENTS  IN  MULTIPLICATION 

37.  The  coefficient  of  a  term  in  a  product  of  two  alge- 
braic expressions  is  the  product  of  the  coefficients  of  the 
given  multiplier  and  multiplicand. 

Thus,  7  X  8a  =  (7  X  8)a  =  56a. 

ab  X  cdm  =  (ah  x  cd)m  =  abcdm. 

EXPONENTS  IN  MULTIPLICATION 

Suppose  we  obtain,  arithmetically,  the  product  of  two 
numbers,  16  and  8.     We  write : 

16 
__8 
128 

16  and  8  are  factors  of  128. 


88  -  MULTIPLICATION- 

38.  The  Factors  of  a  quantity  are  the  quantities  that, 
multiplied  together,  will  produce  it. 

Let  us  now  consider  the  16  and  the  8  as  themselves 
composed  of  factors. 

Then  16  =  2x2x2x2 

and  8  =  2x2x2 

Hence,  128  =  2x2x2x2x2x2x2 

In  the  same  manner : 
27  X  81  =  (3  X  3  X  3)  X  (3  X  3  X  3  X  3). 
36  X  9  X  16  =  (2  X  2  X  3  X  3)  X  (3  X  3)  X  (2  X  2  X  2  X  2) ;  etc. 

It  is  clear,  therefore,  that  all  the  factors  of  a  multipli- 
cand and  a  multiplier  are  included  in  a  product.  But  this 
method  of  expressing  the  fact  is  cumbersome,  and  we  have 
need  for  a  briefer  form.  Hence,  we  use  a  small  number 
at  the  right  and  slightly  above  a  factor  that  occurs  twice 
or  more  in  a  product,  and  our  process  is  Written  thus : 

16  2* 

8       or        28 

128  2^ 

The  product  indicates  in  a  new  way  the  number  of  factors 
composing  it. 

39.  The  product  of  two  or  more  equal  factors  is  called 
a  Power. 

40.  The  number  showing  how  many  times  a  factor 
occurs  in  a  power  is  called  an  Exponent. 

In  common  practice  we  say : 

16  =  2*,  or,  16  is  the  fourth  power  of  2. 
8  =  2^,  or,  8  is  the  third  power  of  2. 
128  =  2\  or,  128  is  the  seventh  power  of  2. 


MULTIPLICATION  OF  MONOMIALS  39 

Similarly,  with  literal  factors  : 

a  X  a^  =  0^+^  =  a^,  which  we  read  "  a  •••  to  the  sixth  power." 
m^  X  m^  =  m^+^  =  m^,  which  we  read  "  a  •••  to  the  ninth  power." 

We  conclude,  therefore,  that : 

41.  The  exponent  of  any  factor  in  a  product  is  the  sum 
of  the  exponents  of  that  factor  in  the  multiplicand  and 
multiplier. 

The  difference  between  a  coefficient  of  a  letter  and  an 
exponent  of  a  letter  must  be  carefully  noted.  A  numeri- 
cal illustration  will  emphasize  the  difference : 

5x2  =  2  +  2  +  2  +  2  +  2  =  10.     Here  5  is  a  coefficient. 
25  =  2x2x2x2x2  =  32.     Here  5  is  an  exponent. 

The  difference  is  obvious. 

The  principle  is  the  same  no  matter  how  many  literal 
factors  occur  in  a  product. 

Thus  :        a%^  x  a%^  =  a%\  a%^c^  x  a^h^c"^  =  a^Wc^- 

x^yz^  X  xyH'^  =  x^y^.         xyz^  x  xyz^  =  x^yh^. 

Note  that  when  no  exponent  is  written  on  a  given  letter,  the 
exponent  is  understood  to  be  "  1."  In  common  practice  a^  is  read 
"a  square";  a^  is  read  "a  cube."  Exponents  of  parentheses  have 
the  same  meaning.     Thus,  (a  +  &)3  =  (a  +  &)(a  +  b)(a  +  &). 

I.    MULTIPLICATION  OF  A  MONOMIAL  BY  A  MONOMIAL 

The  following  illustrates  the  general  process : 
Ex.  1.    Multiply  4  a%^  by  3  aPc^ 

^^2^3  Explanation.     We  first  determine  the  sign 

3  ^^2^2  of  the  result.     Since  the  signs  of  both  monomials 

• are  like,  the  sign  of  the  product  is  positive.    We 

•      12  a%^c^  Result.        ^y^^^   determine  the  coefficient,  which  will   be 


40 


MULTIPLICATION 


3  X  4  =  12.     Then  taking  the  literal  factors  one  at  a  time,  "we  obtain 
for  the  a  factor,  „        ,^- 


ax  a^  =  a^-^^  =  a% 
b^xb^-  62+3  =  b^ 

,2 


for  the  b  factor, 

for  the  c  factor,  c^      (since  this  factor,  occurring  in  the  mul- 

tiplier only,  is  not  changed  in  multiplying). 

Combining  the  sign,  the  coefficient,  and  the  several  powers  of  the 
given  factors,  we  have  the  product, 

+  12  a%^c^ 

It  is  good  practice  to  determine  the  several  parts  of  a 
product  in  this  order : 

1.  The  sign  of  the  product. 

2.  The  coefficient  of  the  product. 

3.  The  exponents  of  the  literal  factors. 


Exercise  8 

Find  the  products  of  the  following : 


1. 

2. 

3.                 4. 

5.                 6. 

7. 

2a 

ba 

^x          -4a 

bm         —5x 

-Sx 

3 

-3 

2x                 a 

-m            -3 

20  X 

8. 

9.                         10. 

11. 

12. 

-5,y 

—  3  a52             4  TY^n 

5a^y 

-%xf 

-14^ 

2a%          -  3  mn 

-xy 
15. 

2xz 

13. 

14. 

16. 

bxyz^ 

—  6  rri^xy 

—  3  mnx 

4.yz^ 

-Zy 

4  c^:i^y 

—  6  nxy 

-12xz^ 

MULTIPLICATION  OF  POLYNOMIALS  41 

17.  Sab  by  2ac.  23.  —mVi/^hy—3mV^, 

18.  17aho  by   -  S  aho.  24.  -lOa^A  by  3  a253^. 

19.  ^253^  by  ^252^,  25.  3  a^Jcc?  by  12  5c%. 

20.  4  3^1/  by   —  7?'y^^.  26.  — 11  cmn^y  by  5  mVa^. 

21.  4w^i  by  —Smny.  27.  —  ISaJ^^^^ljy  5^^^?^, 

22.  12  a2;^3  i^y  _2m%.  28.  15am7i2yby  —  Sam^a;^^. 

II.     MULTIPLICATION  OF  A  POLYNOMIAL  BY  A  POLYNOMIAL 

To  illustrate  the  general  principles  of  multiplication 
when  either  multiplier  or  multiplicand,  or  both,  are  made 
up  of  several  terms,  examine  the  following  : 

If,  arithmetically,  12  is  to  be  multiplied  by  4,  we  obtain 

12 
_4 

48 

Now,  if  we  consider  the  12  as  made  up  of  a  ten  and  two  units,  that 
is,  two  terms,  we  may  make  our  multiplication  in  this  way ; 

10  +  2 
4 

40  +  8  =  48 

In  the  second  multiplication  the  separation  of  the  multiplicand 
produced  two  products,  whose  sum  is  the  same  as  the  product  of  a 
multiplication  in  the  usual  form.  Clearly,  therefore,  we  have  only 
changed  the  form  of  the  arithmetical  process,  and  the  result  is  unchanged. 
In  like  manner : 


(z+    7 
4 

4m   -10 
3m 
12  m2  -  30  m 

5  a6   +15  ac 
3a 
loa2&  +  45a2c 

lOxy   -3 
-    3x 

4a +  28 

-  30  a:2y  +  9  a; 

Each  term  of  the  multiplicand  is  separately  multiplied  by  the 
multiplier. 


By  Separation 

10    +    3 

10    +    2 

100  +  30 

+  20  +  6 

42  MULTIPLICATION 

Suppose,  again,  that  we  multiply  13  by  12. 

By^  the  Arithmetical  Process 

13 

12 

26 
13 
156  100  +  50  +  6  =  156 

In  the  separated  process  the  result  is  the  same  as  in  the  first  pro- 
cess. The  product  of  (10  +  3)  by  10  is  added  to  the  product  of 
(10  +  3)  by  2.  The  only  new  step  is  the  arrangement.  By  beginning 
at  the  left  we  have  brought  the  hundreds,  tens,  and  units  into  proper 
order  for  the  addition  of  the  partial  products. 

And  here,  as  in  the  simpler  case  above. 

Each  term  of  the  multiplicand  is  separately  multiplied  hy 
each  term  of  the  multiplier. 

The  partial  products  are  added  as  in  ordinary  multipli- 
cation. 

Applying  the  same  principle  to  expressions  containing  literal  terms, 
we  have : 

a+4  m  —  5  x  +7 

a+3  m-3  a;-4 


a2  +  4a  m^  -  5m.  x^  +  Ta; 

+  3  g  +  12  -  3  m  +  15  -  4  a:  -  28 

a2  +  7  a  +  12  m^  -  8  w  +  15  x^  +  3  x  -  28 

Explanation  of  the  First  Multiplication 

For  the  first  partial  product :  +ax  +  a  =  +  a2,  +ax+4  =  +  4a. 

Hence,  the  first  partial  product,  d^  -^  4:a. 

For  the  second  partial  product :    +3x+a=  +  3a,  +3x+4=  +  12. 

Hence,  the  second  partial  product,   +  3  a  +  12. 

The  first  term  of  the  second  partial  product  is  written  under  the 
second  term  of  the  multiplier,  thus  bringing  like  powers  of  a  in  the 
same  column.  The  addition  of  the  partial  products  gives  the  result, 
a2  +  7  a  +  12. 


MULTIPLICATION  OF  POLYNOMIALS  43 

In  this  multiplication  the  signs  were  all  plus  because  of  like  signs 
in  every  multiplication  of  coefficients.  In  tlie  other  two  examples 
both  plus  and  minus  signs  occur,  and  the  solution  of  each  should  be 
carefully  gone  over  before  the  student  attempts  to  apply  the  principles 
involved. 

42.  Arrangement  of  the  Terms  of  Polynomials.  In  mul- 
tiplying polynomials  it  is  convenient  to  arrange  the  terms 
of  both  multiplicand  and  multiplier  so  that  like  terms  of 
the  partial  products  shall  fall  in  the  same  column.  This 
arrangement  is  made  by  writing  both  expressions  in  ac- 
cordance with  the  powers  of  some  letter  occurring  in  each 
of  them. 

When  the  powers  of  a  letter  in  an  expression  increase 
from  left  to  right,  as  in 

a  +  a^  +  a^-ha^  +  a^, 

the  expression  is  said  to  be  arranged  in  the  ascending 
powers  of  a. 

When,  on  the  other  hand,  the  powers  of  the  selected 
letter  decrease  from  left  to  right,  as  in 

a"^  +  a5  +  aS  +  a2  4-  a, 

the  expression  is  said  to  be  arranged  in  the  descending 
powers  of  a. 

43.  The  Degree  of  an  Algebraic  Expression.  That  par- 
ticular term  whose  power  is  the  highest  of  any  in  a  given 
expression  determines  the  degree  of  the  expression.    Thus  : 

a^  +  a  +  1  is  an  expression  of  the  second  degree. 
x^  —  x^  —  x^  is  an  expression  of  the  fourth  degree. 
x'^y  —  x^y^  +  7  is  an  expression  of  the  fifth  degree,  since  its  highest 
power  is  a  product  of  a  first  and  a  fourth  power. 


44  MULTIPLICATION 

A  numerical  factor  in  a  term  is  not  considered  in  naming 
its  degree. 

From  the  principles  of  exponents,  Arte  41,  the  degree 
of  the  product  of  two  algebraic  expressions  is  equal  to  the 
sum  of  the  degrees  of  the  given  expressions. 

We  will  now  state  the  general  process  for  multiplication 
of  polynomials. 

44.   To  multiply  a  polynomial  by  a  polynomial. 

Arrange  the  terms  of  each  polynomial  according  to  the 
ascending  or  descending  powers  of  the  same  letter. 

Multiply  all  the  terms  of  the  multiplicand  hy  each  term  of 
the  multiplier^  observing  the  principles  that  govern  the  signs. 

Add  the  partial  products  thus  formed. 


Exercise  9 

Find  the  products  of  the  following : 

1.                             2.                                  3. 

a  +  55             2a2_3a                bah-^c 
a                     4ta                      —  2a 

4. 

a2_2a  +  3 
a2 

5.  3  a +  17  by  2  a. 

6.  VI  ah -10  by  3a5. 

7.  8  a^m  —  3  a2^  by  5  amn, 

8.  16 xy —10 xz—1  yz  by  —  2 xyz. 

9.  _l0a8_3^2+2a-4  by  -6a2. 
10.  a8-3a26  +  3a62-53  by  -a%\ 


MULTIPLICATION  OF  POLYNOMIALS  46 


11. 

12. 

13. 

2a  +    3                            5a2  + 

7 

3a  +    4a: 

3a  -    7                            2a2- 

11 

3a   -    4a: 

6a2+    9a                ■       10a* +  14 

a2 

9  a2  +  12  ax 

-14a -21 

■55 

ab- 

77                      -  12  ax  -  16  a;2 

6a2-    5a-21               10  a* - 

-41 

as- 

77              9  a2               -  16  x^ 

14.   2a  +  S  hj  a  +  1. 

ia. 

2m- S  by  Sm  +  d, 

15.   Sx-2  by  2a;  +  l. 

20. 

4^/2  +  5  by  5^2_i2. 

16.   2a:+3  by  ir-3. 

21. 

5ah  —  c  by  4  a6  —  3  <?. 

17.   42;-l  by  2  a; +  3. 

22. 

5a;3_l  by  11  a;^  -  2. 

18.    3a- 7  by  2a- 6. 

23. 

Sab^-2hc  by  2ah'^-5hc. 

24.  16a^-10  by  5a^  +  ll. 

25.  4  mwa;2  _  3  ji^^  by  2  mwa;^  +  5  tio;^. 

26.  a3  +  3a24-3a  +  l  by  a^+2a  +  l. 

a8+3a2+    3a   +    1 
a2  +  2a   +    1 


a6  +  3a*+    3a8+       a^ 

2a*+    6a8+    6a2  +  2a 

a8+    3aS  +  3a+l 

o5  +  5  a*  +  10  a8  +  10  a2  +  5  a  +  1 

27.  a2  +  2a  +  l  by  a2  4-2a  +  l. 

28.  w3-3m24- 3m-l  by  m^-2m-^l. 

29.  2ic2+3a:  +  l  by  a^-x  +  l. 

30.  5^3_3^2_|.2y_4  by  «/2_4y +  2. 

31.  a3  -  a25  +  a62 _  53  by  a24-^^'  +  ^'2. 

32.  m^  +  m^  +  m^  +  m  +  l  by  w  — 1. 


46  MULTIPLICATION 

33.  27  -{-lSa  +  12a^  +  Sa^  hj  S-2a. 

34.  l-2x-h4:x'^-Sx^-\-163^  by  l  +  2rc. 

35.  2a-{- a^ -Sa^ -1  hy  2- a  +  a^,    ■ 

First  arrange  the  terms  so  as  to  have  the  same  order  in  both  mul- 
tiplicand and  m.ultiplier.     In  descending  order  we  write : 

a^-Sa'^-\-2a-\ 
a^-     g  +2 

(The  student  will  complete  the  multiplication.) 

36.  a  +  l  —  a^  +  a^  hy  a-'2-\-a^. 

37.  a^-x^-\-6-Sa^-x  by  x^-{-4:-^a^-2x. 

38.  'lO-5x-\-Sx^-x^  by  x  -  5x^- 2a^ -{-a^. 

39.  4a2_  8^3- 3^  +  16  4- a*  by  2a-  8-6^2  + a3. 

40.  a  +  b  +  2  by  a  +  b-2. 

a  +      6  +  2 
a  +      b-2 


a2  +    a&  +  2  a 

+     a6  +  62  +  2  6 

-2a         -26-4 

a2  +  2  a6  +  6^  -  4 

41.   a2  _|_  52  _^  ^  _,_  2  «5  _  ^c  _  j^  by  a  +  b-\-c. 

First  arrange  terms  according  to  the  descending  powers  of  a. 

a^-h  2ab-  ac+      W-      bc-{-   c^ 

a  +       b+     c 

a8  +  2  a%  -  a^c  +     ab^  -     abc  +  ac^ 

+     a%  +2a62-     abc  +¥-bh+hc'^ 

+aV -f  2  q6c  -  ac^        +  6^0  -  bc^  +  c^ 

a«  +  3a26  +  3  n62  +  6»  +0^ 


MULTIPLICATION  OF  POLYNOMIALS  47 

42.  7W  +  w  -t-  4  by  w  +  7^  —  4. 

43.  X  -\-y  -{-z  by  x-i-  ^  +  z. 

44.  m^+2  mn  -^n^  -\-x  by  m^  +  2  w/i  +  w^  —  a?. 

45.  x^ -\-  y^ -\- a^  -\- 2  xy  —  ax  —  ay  by  a;  +  3/  +  a. 

46.  aH-6  +  (?H-c?bya  —  6— <?  —  c?. 

Multiplication  of  polynomials  is  frequently  indicated 
by  the  use  of  parentheses.  In  such  indicated  operations 
any  two  of  the  given  expressions  are  multiplied  and  their 
product  is  multiplied  by  the  third,  etc.  An  exponent  upon 
a  parenthesis  indicates  the  multiplication  of  the  given 
expression  by  itself  as  many  times  as  the  number  of  the 
exponent. 

47.   (a -2)3.  48.   (a  +  5)(a4-6)(a-3). 

a  -2 
a  -2 


I- 


a2-2a 

-2a  + 

4 

a2-4a  + 

4 

a  -2 

a8-4a2  + 

4a 

-2a2  + 

8a- 

-8 

a  +    5 

a  +   6 

a2-i-   5  a 

+   6a  +30 

a2+lla  +30 

a  -   3 

a8  +  11  a2  +  30  a 

-   3a2-33a- 

-90 

a8_6a2  +  12a-8  0^+    8a2-   3a-90 

49.  (a  4- 2)2.  51.    (2  m- 3^1)3.      53.    (Za^-bxy^. 

50.  (a-3:r)2.       52.    (4^2-5  2^)2.      54.    (b aV^ -1  ahcf. 

55.  (m2 -f- m  —  l)(m2  — m  —  l)(w2— 1). 

56.  (^x'^J^x  +  V)(x^-x  +  V)Qi^-x'^-\-V), 

57.  (^+-5)(a?-3)(a;-5)(:r  +  3). 


48  MULTIPLICATION 

When  the  terms  of  the  multiplicand  and  multiplier  are 
dissimilar,  the  partial  products  cannot  be  added.  In  such 
cases  the  additions  of  the  partial  products  can  only  be 
indicated. 

58.  (a  +  a;)(a  +  ^). 

a  +  X 
a  -h  y 
a^  +  ax 

+  ay-\-xy 
a^  -^ax^  ay  +  xy 

59.  (a  +  5)(a  +  c).  63.  (^a  +  7n)(h-{-n). 

60.  (2rz;  +  m)(3a^+2w).  64.  (2a  +  3  J)(3c?  -  4(^). 

61.  (ah-c){ah-d).  65.  (a^  ^-V)(jj(P'- x). 

62.  (2m-\-n')(bm  —  y'),  66.  (a-\-h-{-V)(^a  —  x), 

Perform  indicated  operations  and  simplify : 

67.  3a2-(a-2)(3a  +  l)-5a. 

68.  (5 -1)2 +(6 +  1)2 -2(52+1). 

69.  (4a-l)(a  +  2)-(2a-3)(2a  +  5). 

70.  (a  +  3)2-2a(a  +  4)  +  (a-2)(a  +  4). 

71.  rc(a;-l)+2:(2:  +  l)(rc+2)-ic2(a;  +  4). 

72.  m(jn-\-n—l)—n(m—n  +  l)  —  m(m--V), 

73.  c(^  +  l)_[(e-l)2_(l-e)]. 

74.  (a_5_l)2  +  a(5  +  l-a)  +  5(a-6-l). 


CHAPTER  V 
DIVISION.     REVIEW 

45.  Division  is  the  process  of  finding  one  of  two  factors 
when  their  product  and  the  other  factor  are  given. 

The  Dividend  is  the  given  product. 

The  Divisor  is  the  given  factor. 

The  Quotient  is  the  factor  to  be  found. 

Division  is  the  inverse  of  multiplication,  and  upon  this 
fact  we  base  the  principles  by  which  we  obtain  the  signs, 
coefficients,  and  exponents  of  our  quotients. 

SIGNS  IN  DIVISION 

« 

From  Arts.  35  and  36  we  obtain  : 

(+a)x(+^)=  +  «^  +ah-^-\-a  =  +  b     (1) 

Hence, 
(+a)x(-^>)=-a5     inversely,      -ab-t-  +  a=-h     (2) 

(_a)x(+^>)=-a5         ^y  -ah-^-a=  +  h     (3) 

division, 
(-a)x(-5)=  +  a6  -{-ah^  —a=—h     (4) 

Therefore,  we  conclude : 
From  (1)  and  (3): 

46.  Like  signs  in  division  give  + . 

From  (2)  and  (4): 

47.  Unlike  signs  in  division  give  —  • 

p.  H.  S.  FIRST  TEAR  ALG. 4         49 


50  DIVISION.      BE  VIEW 

COEFFICIENTS  IN  DIVISION 

Since  two  factors  multiplied  together  give  a  product,  it 
follows  that  this  product  divided  by  either  of  the  factors 
will  give  the  other  factor.     Hence  : 

48.  The  coefficient  of  a  quotient  is  obtained  by  dividing 
the  coefficient  of  the  dividend  by  the  coefficient  of  the  divisor, 

EXPONENTS  IN  DIVISION 

By  Art.  41, 

This  might  be  written, 

aaaa  x  aa  =  aaaaaa. 

Now,  if  the  product  of  six  factors  be  divided  by  the  group  of  two 
factors,  we  obtain  by  cancellation. 

Or,  subtracting  the  number  of  a  factors  in  the  divisor  from  the 
number  of  a  factors  in  the  dividend,  we  obtain  the  number  of  a  factors 
in  the  quotient.     Hence : 

Similarly,  a^^a'^  =  a^-^  =  a*. 

x^^x*  =  x^-^  =  x^. 

From  which  we  state  the  general  principle  for  the  expo- 
nents of  quotients : 

49.  The  exponent  of  a  factor  in  a  quotient  is  the  differ- 
ence between  the  exponents  of  that  factor  in  the  dividend  and 
divisor. 


DIVISION  OF  MONOMIALS  51 

It  is  good  practice  to  obtain  the  several  parts  of  a  quo- 
tient in, this  order  ; 

1.  The  sign  of  the  quotient. 

2.  The  coefficient  of  the  quotient. 

3.  The  exponents  of  the  literal  factors. 

I.    DIVISION  OF  A  MONOMIAL  BY  A  MONOMIAL 

The  general  process  of  dividing  a  monomial  by  a  mono- 
mial is  as  follows : 

Ex.  1.   Divide  10  a^  by  5  aK 

5  a8)10a5        Explanation.    ,(1)  Since  the  signs  of  the  coefficients 
2  a^    are  like,  the  sign  of  the  quotient  will  be  + .     (2)  Divid- 
ing the  given  coefficients,  we  have  10-4-5  =  2,  the  coefficient 
of  the  quotient.     (3)  For  the  exponent  of  the  literal  factor,  a^  -^  a^= 
a^-s  =  a^.    Hence,  the  complete  quotient,  2  a^. 

Ex.   2.   Divide  36  a^a^  by  -  12  aV. 

—  12  a%8)36  a^x^  Explanation.  (1)  Since  the  signs  of  the  coeffi- 
3  a^x  cients  are  unlike,  the  sign  of  the  quotient  will  be 
— .  (2)  Dividing  the  given  coefficients,  we  have 
36  -4- 12  =  3,  the  coefficient  of  the  quotient.  (3)  For  the  exponents  of 
the  literal  factors,  a^  ^  a^  =  a^  and  x^  -^  x^  =  x.  Hence,  the  complete 
quotient,  —  3  a^x.  From  these  illustrations  we  state  the  general  pro- 
cess for  the  division  of  monomials. 

50.   To  divide  a  monomial  by  a  monomial : 

Determine  the  sign  of  the  quotient^  +  or  —  according  as 
the  signs  of  the  dividend  and  divisor  are  like  or  unlike. 

Divide  the  coefficient  of  the  dividend  hy  the  coefficient  of 
the  divisor  and  affix  the  sign  obtained. 

Annex  the  literal  factors,  giving  to  each  an  exponent  equal 
to  the  difference  between  its  exponent  in  the  dividend  and  its 
exponent  in  the  divisor. 


52  DIVISION.      REVIEW 

E2:ercise  10 

Obtain  the  quotients  in  the  following : 

1.  2.  3.  4.  5. 

6.  7.  8. 

5a)15a3  -8^2)24^8  Ibxy)-^^ 

9.  10. 

-8^25) -16^352  Zm^y)-'nrY&y^ 

11.  12.  13.  14. 

5:i;2^  -18a;2«/2a  -25m2  -3a26c?(^ 

II.    DIVISION  OP  A  POLYNOMIAL  BY  A  MONOMIAL 

A  simple  numerical  example  will  illustrate  the  principle 
of  division  upon  which  we  base  our  work  when  the  divi- 
dend is  a  polynomial  and  the  divisor  a  monomial. 

Suppose  we  divide  48  by  4. 

^4?    or,  by  separation,    ^M  +  l      ^„ 
12        '    J'      ^  10  +  2  =  12. 

Simil  rl  ^  a2^15a^-_12af  ~  5 a)10a  -  15a2+ 20  gS 

imiary,  5a2-    4a  -  2    +    3  a  -     4  a2 

Hence,  the  general  process. 

51.   To  divide  a  polynomial  by  a  monomial : 

Divide  each  term  of  the  dividend  hy  the  divisor  and  con- 
nect the  several  quotients  hy  the  proper  signs. 


DIVISION  OF  POLYNOMIALS  53 

Exercise  11 

Obtain  the  quotients  in  the  following : 

1.  2.  3. 

a)a±5ab  4  a)Sa^-12a^  -  2  a)10  a^  +  6 ac 

4.  5. 

6. 

7.  7a3_21a2  by  la. 

8.  4  m^o:  —  12  wa;2  by  4  mx, 

9.  —  25  w%2  +  20  mW  —  15  mhi^  by  5  wV. 

10.  5  a%- 10  aH^ +  15  ah^  hy   -bah. 

11.  -18^6-24^5 4-30m*-36m34-42w2  by  -6w. 

III.    DIVISION  OP  A  POLYNOMIAL  BY  A  POLYNOMIAL 

When  both  dividend  and  divisor  are  polynomials,  the 
principles  already  used  are  somewhat  extended.  We  will 
illustrate  by  a  numerical  example.  Suppose  we  divide 
156  by  12. 

12)156(13  ,        .  100  +  50  +  6(10  +  2 

j2  or,  separating  both  dm-  loo  +  20  10  +  3  =  13 

~^  dend    and    divisor    as  on  ,  e 

og  in  former  divisions,  SO  4-  6 

In  the  second  process  the  divisor  is  written  at  the  right  of  the  divi- 
dend.    The  process  in  its  successive  steps  is  as  follows : 

First.    100  -f-  10  =  10,  the  first  term  of  the  quotient. 


64  DIVISION.      REVIEW 

Second.  Multiplying  (10  +  2)  by  10,  we  obtain  (100  +  20),  which 
product  is  subtracted  from  the  given  dividend.  The  remainder, 
(30  +  6),  is  the  new  dividend. 

Third.  30  -=-  10  =  3,  the  second  term  of  the  quotient.  Repeating 
the  process  of  multiplication  and  subtraction,  the  remainder  is  zero. 
Hence,  the  quotient  is  (10  +  3). 

Similarly,  Dividend  a^  -f  5  a  +  6  (a  +  2  Divisor. 
a2  +  2  a  a  +  3  Quotient. 

+  3a+  6 
+  3a  +  6 

First,    a^  -^  a  =  a,  the  first  term  of  the  quotient. 

Second.  Multiplying  (a  +  2)  by  a,  we  obtain  (a^  +  2a),  which 
product  is  subtracted  from  the  given  dividend.  The  remainder, 
(3  a  +  6),  is  the  new  dividend. 

Third.     3  a  -^  a  =  3,  the  second  term  of  the  quotient. 

Fourth.  Multiplying  (a  +  2)  by  3,  we  obtain  (3  a +  6),  which, 
subtracted  from  the  last  dividend,  gives  a  remainder  of  zero,  and 
the  division  is  complete. 

And,  in  a  more  difficult  case,  the  process  is  identical : 

Dividend  a*  +     a^  _  5  ^^2  _,.  13  ^  _  6  (gg  -  2  a  +  3  Divisor. 
a4  _  2  a8  +  3  a2  a^  +  3  a  ^  2  Quotient. 


+  3a8 
+  3a8 

-8a2 
-6a2 

+  13  a 
+    9a 

-2a2 
-2a2 

4a- 
4a- 

-6 
-6 

Hence,  the  general  process  for  division  of  polynomials. 

52.   To  divide  a  polynomial  by  a  polynomial : 

Arrange  the  dividend  and  the  divisor  according  to  the 
ascending  or  descending  powers  of  some  common  letter. 

Divide  the  first  term  of  the  dividend  hy  the  first  term  of 
the  divisor. 


DIVISION  OF  POLYNOMIALS  55 

Write  the  result  for  the  first  term  of  the  quotient. 

Multiply  all  the  terms  of  the  divisor  hy  the  first  term  of 
the  quotient  and  subtract  the  product  from  the  dividend. 

The  remainder^  if  any,  is  a  new  dividend,  and  the  opera- 
tion is  repeated  with  successive  dividends  until  the  remainder 
becomes  zero. 

Exercise  12 

Divide : 

1.  2m2+llm-f5  by  2^  +  1. 

2.  2x^-\-x-Q  by  2x-Z. 

3.  3  c2  +  5  (?  -  12  by  (?  +  3. 

4.  3m2  +  14w  +  15  by  3m +  5. 

5.  16  +  8a:  +  rz;2  by  4  +  2;. 

6.  x^-^x-12  by  x-%. 

7.  7y2  +  i23«/-54  by  7?/ -3. 

8.  ?>a%''--\-l\ab-20  by  3a6-4. 

9.  21  m%*  +  26  m2/?,2  _  15  by  7mV-S. 

10.  10a*  +  23a2^-21a;*  by  lOa^-la^. 

11.  a^-\-Sa^-\-Sx-^l  by  x+1. 

12.  m^  +  m2  —  3  m  —  6  by  m  —  2. 

13.  a3  -  6  a2  +  8  a  -  15  by  a  -  5. 

14.  mV  —  ■m^/i*  —  7  ^2^2  +  3  by  mV  —  3. 

15.  1  +  a  -  3  ^2  4-  9  ^3  by  1  +  3  a. 

16.  a*  +  2  a3  +  2  a2  +  a  -  6  by  a2  +  a  -  2. 

17.  2x^-a^-Sx^-\-5x-2  by  22;2-3a;  +  2. 


56  DIVISION.      REVIEW 

18.  *9a+10-22a2  +  2a*  +  a3  by  2a2-l-a. 

19.  9a^-6x+llx^+6a^-10  by  -2  +  32^. 

20.  2  m^  -  7  +  8  w  +  8  w*  +  9  m2  by  -  1  +  2  m^  +  m. 

21.  6a^-{-2a^-5a^-6-\-ba-lla^  by  2  +  2a2_a. 

22.  -  a*  +  a2  _  2  a3  -  1  4-  a^  +  2  a^  by  a2  +  1  -  a  +  a3. 

23.   a;2-      y\xj-j_ 
x^  —  xy     X  -\-  y 
+  xy  -y^ 

-\-xy  -y'^ 


24.   x8  -       ?/8(x 


a;8  —  x^y      x^  +  xy  -\-  y^ 

+  a:V 

+  x'^y  -  xy^ 


25.  a*  —         m\a  -\-  m 


a*  +  ahn 

a8- 

a2m4- 

am^- 

—  a^m 

—  ahn 

-ahn^ 

+  ahn^ 

+  ahn^  +  am* 

—  am^  ■ 

-m* 

—  am^  ■ 

-m* 

26.  a2  _  52  ijy  ^  ^  5,  28.    a^  -  53  by  a  -  6. 

27.  a^  —  h^  by  a—  h,  29.    a*  —  5*  by  a  +  ^. 
*  Arrange  terms  of  dividend  and  divisor  in  descending  order. 


DIVISION  OF  POL  TN0MIAL8  57 

30.  a3  -  8  by  a  -  2.  32.    8  a^  +  27  5^  by  2  a  +  3  h. 

31.  m^  +  32  by  m  +  2.  33.    16  m*  -  1  by  2  w  +  1. 

34.  125  a656  + 27^3  by  5^262 +  3^. 

35.  81  a:*/  -  16  ^4  by  3  a:?/  +  2  2. 

36.  a8-3a&c+    S^  +  c8(a  +    5+    c 

a3  +     ^2^  ^  a%         (a2  _  a6  -  ac  +  &2  _  &c  +  c« 

-  a26-a2c.-3a6c+       b^-\-    c^ 

-  a%-ab^-     abc 


-ah  + 
-ah 

ab^- 

2abc+  b^  +  c^ 
abc  -  ac'^ 

+ 
+ 

ab^- 
ab^ 

abc  +  ac"^ -^^  b^  +  c^ 
+  b^  +  bh 

: 

abc  +  ac"^         - 
abc                   — 

bh+  c« 

62c  -  &c2 

^ac"^ 
+  ac^ 

In  this  division  the  student  will  note  that  in  all  new  dividends  the 
letter  a  is  given  precedence,  with  powers  of  b  and  c  following  in  their 
original  order. 

37.  a^  +  ^ah  +  y^-c^  hj  a-\-h  +  c, 

38.  a;2  —  4  a;  H-  4  —  ?/2  by  a:  —  2  —  y. 

39.  c^  +  2cd  +  d?-4:x^  hy  c-\-d—2x, 

40.  4  2^  +  12  a:  +  9  -  ^2  by  2  a;  +  3  +  ^. 

41.  ^2  +  &2  4.  c2  +  2  ^6  +  2  a(?  +  2  5c?  by  a  4-  J  +  <?. 

42.  ofi  -{-  y^ -{■  z^  —  2>  xyz  hj  x  -{-  y  +  z. 

43.  w^  —  2  m^  +  1  by  w2  _  2  m  +  1. 


58  DIVISION,      BEVIEW 

GENERAL   REVIEW  — GROUP   II 

The  following  group  of  principles  presented  in  an 
abbreviated  form  covers  the  new  work  since  the  preced- 
ing group.  It  is  suggested  that  the  student  review  again 
the  principles  covered  in  the  first  group,  for  the  elementary 
processes  that  have  been  thus  far  presented  cannot  be  too 
well  fixed  before  more  advanced  work  is  attempted. 

SUBTRACTION 

The  subtrahend^  with  all  signs  changed^  is  added  to  the 
minuend. 

MULTIPLICATION 

Signs.  Like  signs  in  multiplicand  and  multiplier  produce 
a  +  product.  Unlike  signs  in  multiplicand  and  multiplier 
produce  a  —  product. 

Exponents.  The  exponent  of  any  factor  in  a  product  is 
the  sum  of  the  exponents  of  that  factor  in  the  multiplicand 
and  multiplier. 

Arrangement  of  Terms.  Multiplicand  and  multiplier 
should  he  arranged  in  the  same  order^  both  ascending  or  both 
descending. 

DIVISION 

Signs.  Like  signs  in  dividend  and  divisor  produce  a  + 
quotient.  Unlike  signs  in  dividend  and  divisor  produce 
a  —  quotient. 

Exponents.  The  exponent  of  a  factor  iyi  a  quotient  is  the 
difference  between  the  exponents  of  that  factor  in  the  dividend 
and  the  divisor. 

Arrangement  of  Terms.  Dividend  and  divisor  should 
be  arranged  in  the  same  order^  both  ascending  or  both  de- 
scending. 


REVIEW  59 

Exercise  13 
REVIEW 

1.  Add  (a  +  2)(a  +  3)  and  (a  +  1)2. 

2.  What  is  the  sum  of  3(a2  _  ^  +  2)  -  2(a  -  3  +  a2)  - 
(«-!)(« +  3)? 

3.  Subtract  (a  +  2)2  from  (a  +  1)^. 

4.  Subtract  (a  +  2)(a  —  3)  from  a^  4.  ^  _  5. 

5.  By   how   much   does  4(a  —  3)  —  3(5  +  1)    exceed 
2(a  +  6  -  2)  ? 

6.  Subtract  (a  +  3)  from  (a  —  2)2  and  multiply  the 
result  by  (a—  1). 

7.  From  (a2+  3)2  take  (a+  3)3. 

8.  By  how  much  does  (a+ 6 +  (?)2  exceed  2(aJ  +  «c4-5(?)? 

9.  What  must  be  added  to  (a  +  a;+  1)^  that  the  sum 
may  be  (^a-\-x—  1)2  ? 

10.  Subtract  (a  +  3)  from  (a2  —  2  a)  and  from  the  re- 
mainder take  «2  _  4  ^  _  3. 

11.  Prove  that  w(a  —  rr)  +  a(rc  —  m)  +  a;(m  —  a)  =  0. 

12.  Multiply  the  sum  of  a^  -j.  2  and  a^-\-a  —  l  by  d^  —  1. 

13.  Multiply  the   sum  of    (a  +  2)   and   («2  _  3  ^)   by 
[«(a  +  l)_(2a-2)]. 

14.  Simplify  a  +  5  -  [(3  a  -  5)  +  (a  -  3  6)  -  a]  +  [a  - 
(5  —  2  a)]  and  multiply  the  result  by  a  —  4  5. 

15.  Simplify  (2;  +  l)(a:+3)(rr  +  4)-(rr-l)(a;-3)(a:-4). 

16.  Simplify  (c  +  l)c-  [(c- 1)2- (1  -  c)]. 

17.  Simplify  (a  +  a;)  (a  +  y)  —  ((X  —  a:)  (<^  —  y). 


60  DIVISION.      REVIEW 

18.  Divide  a^  +  3  ^2  +  3  a  +  1  by  (a  +  1)(«  +  1). 

19.  Find  the  continued  product  of  (2a  +  3)(3a— 2) 
(a-1). 

20.  Simplify  3a-[55-(2aH-J35-(a-[5-^^=^2T])l)], 
and  multiply  the  result  by  (a  +  3  6). 

21.  Find  the  dividend  if  the  quotient  is  a  —  2  and  the 
divisor  a^  —  a-\-Q, 

22.  The  product  of  two  expressions  is  a*  +  11  a^—  12  a 
—  5  a^  +  6,  and  one  of  them  is  3  —  3  a  +  a^.  Find  the 
other  expression. 

23.  The  difference  between  two  expressions  is  a^—^x 
+  2,  and  the  greater  expression  is  4iX^  —  2x-{-l.  Find 
the  smaller  expression. 

24.  Divide  the  product  of  (a^— 4)(a2— 9)  by  a^-f-a— 6. 

25.  Subtract  a^— 4a;  from  (x^-\-x-{-l')(x-\-2~)  and  divide 
the  result  by  (3  a;  +  1). 

26.  Add  a^-\-a-2>  to  the  quotient  of  Qa^-\-^a^-a-\-12') 
^(a  +  4). 

27.  Divide  [(a2  -  2  a  -  l)(a  -  1)  +  2  a2  _  2  a]  by  [(a 
+  2)(a  +  l)-(a2+2a  +  3)]. 

28.  What   is   the    sum   of    [(a  + 1)^— «(a  + 3)]    and 

29.  Subtract  a—  [2a  —  (  —  a—  l-f-a)]  from  1+  { —  a  — 
[-(-2a  +  l)]i. 

30.  Add  the  quotients  of  [(a^  —  1) -?- (^  —  1)]  and  [(a^ 
_2a;  +  l)-j-(a;-l)]. 


CHAPTER  VI 
THE   SIMPLE   EQUATION 

53.  An  Equation  is  an  expression  of  equality  between 
two  quantities. 

Illustration.  If  five  books  of  exactly  the  same  size  and 
kind  weigh  all  together  ten  pounds,  we  have  a  known  and 
an  unknown  quantity.     For, 

The  weight  of  the  whole  number  is  known. 

The  weight  of  one  book  is  unknown. 

But  the  conditions  given  permit  us  to  write  the  fact 

that,  in  weight,         i-  ^      ^        ^r^  i 

^  o  books  =  10  pounds. 

Or,  if  "  h  "  stand  for  the  weight  of  one  book,  we  abbre- 
viate thus  :  r  r        -if. 

0  0  =  lU. 

And  we  have  an  equation  between  two  quantities,  —  a 
certain  number  of  books  on  one  hand,  and  a  certain  number 
of  pounds  on  the  other. 

Now  if  5  books  weigh  ten  pounds,  1  book  weighs  (10  h- 5) 
pounds,  or  2  pounds.     Or,  from  our  equation, 

5  =  2. 
Hence,  our  conclusion. 

54.  From  an  equation  we  obtain  the  value  of  an  unknown 
quantity/  in  terms  of  a  given  known  quantity/. 

61 


62  THE  SIMPLE  EQUATION 

55.  The  expression  obtained  for  the  value  of  an  unknown 
quantity/  in  an  equation  is  called  a  Moot. 

Thus :  2  is  a  root  of  the  equation  5  &  =  10. 

56.  The  Left  Member  of  an  equation  is  the  expression  at 
the  left  of  the  sign  of  equality.  The  Right  Member  is  the 
expression  at  the  right  of  the  sign  of  equality. 

57.  A  Simple  Equation  is  an  equation  which,  when  re- 
duced to  its  simplest  form,  has  no  power  of  the  unknown 
quantity  higher  than  the  first  power.  The  following  are 
examples  of  simple  equations  : 

5  a?  =  15  is  an  equation  in  x. 
7  y  =  35  is  an  equation  in  y. 
5  2  =  40  is  an  equation  in  z. 

3  a;  +  2  =  2  a;  -i-  7  is  an  equation  in  x,  but  is  not  reduced. 
(x  +  5)2  =  a:^  -f  7  a;  +  6  is  an  equation  in  x  that,  when  simplified, 
will  contain  only  the  first  power  of  x. 

The  final  letters  of  the  alphabet  are  commonly  used  for 
unknown  quantities  in  equations. 

58.  The  principles  upon  which  the  solution  of  equa- 
tions depends  are  the  truths  called 

AXIOMS 

1.  If  equals  are  added  to  equals^  the  sums  are  equal. 

2.  If  equals  are  subtracted  from  equals,  the  remainders 
are  equal. 

3.  If  equals  are  multiplied  by  equals,  the  products  are 
equal. 

4.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 

Applications  of  these  axioms  to  processes  with  equations 
may  be  illustrated  by  the  following  numerical  operations : 


TRANSPOSITION  OF  TERMS  IN  EQUATIONS         63 

12  =  10  +  2  12  =  10  +  2 

By  Ax.  1  add     2=     2   By  Ax.  2  subtract  2=     2 

14  =  10  +  4  (1)  10=  10    (2) 

12  =  10  +  2  12  =  10  +  2 

By  Ax.  3  multiply  _2 2        By  Ax.  4  divide      ^  =  \^-  +  f 

24  =  20  +  4  (3)  6  =  5  +  1     (4) 

From  which  we  conclude  : 

From  (1)  and  (2).      77ie  same  quantity/  may  he  added  to,  or 

subtracted  from^  both  members  of  an 

equation. 
From  (3)  and  (4).     Both  members  of  an  equation  may  he 

multiplied  or  divided  by  the  same 

quantity. 

Or,  taking  the  generral  equation,  A  =  B. 

From  (1)  A+C  =  B+C.  From  (3)  AC  =  BC. 

From  (2)  A-C  =  B-C.  From  (4)  A---C  =  B^C. 

TRANSPOSITION  OF  TERMS  IN  EQUATIONS 

Frequently  we  are  required  to  find  the  root  of,  or  solve^ 
equations  in  which  the  known  quantity  occurs  in  both 
members.     Thus :  5  a;  +  2  =  17 

We  want  only  the  x  term  in  the  left  member,  so  we  will 
subtract  2  from  that  member.  But  if  we  subtract  a  num- 
ber from  one  member,  we  must  subtract  the  same  number 
from  the  other  member  also.     That  is, 

5  re  +  2  =  17 

By  Ax.  2,  Subtracting :  2=2 

5  a;  =  17  -  2 


64  THE  SIMPLE  EQUATION 

This  is  the  same  result  that  would  have  followed  from 
transposing  the  2  to  the  right  member  and  changing  its 
sign..     Hence, 

59.  A  term  may  he  transposed  from  one  member  of  an 
equation  to  the  other  member  by  changing  its  sign  from  +  to 
— ,  or  from  —  to  -\-, 

We  are  now  ready  for  the 

GENERAL  METHOD  FOR  SOLVING  SIMPLE  EQUATIONS 

60.  To  solve  a  simple  equation : 

Perform  all  indicated  multiplications  and  remove  all 
parentheses. 

Transpose  the  terms  containing  the  unknown  quantity  to 
the  left  member^  and  all  other  terms  to  the  right  member  of 
the  equation. 

Collect  the  terms  in  each  member. 

Divide  both  members  by  the  coefficient  of  the  unknown 
quantity. 

In  the  solution  of  equations  the  following  will  frequently 
save  much  labor: 

(1)  The  same  term  with  the  same  sign  in  both  members  of 
an  equation  may  be  canceled. 

(2)  All  the  signs  of  the  terms  of  an  equation  may  be 
changed  without  destroying  the  equality. 

Exercise  14 

Solve  ths  following  equations : 
1.  5a;=35.     2.  7a;=-56,     3.  -4a:=20,     4.  -3a:=-27, 


METHOD  FOR   SOLVING   SIMPLE  EQUATIONS         ^6 

5.  3^  =  5,  6.    -6a:  =  19,  7.  13a:=0, 

x  =  ^.  X—  —  ^g^.  2;  =  0. 

8.  3a;  =  18.  14.    -18a;  =  54.       20.    -5^=16. 

9.  7a:  =  42.  15.    -52:=-20.      21.   3a:=-13. 

10.  5:c=90.  16.  -3  a;  =-39.  22.  -2  2;  =  -9. 

11.  32;  =  -18.  17.  42^=1.  23.  3a;  =  0. 

12.  62;= -30.  18.  92^=5.  24.  72;=  0. 

13.  -3a:=12.  19.  5rr=13.  25.  -8a:=0. 

26.  Solve  5:^  +  2=17,  27.   Solve  7a:- 3  =  3a:+ 13, 
5x=17-2,  7ar-3a:=13  +  3, 

5a;  =  15,  4x=  16,  - 

a;  =  3.     Result.  a;  =  4.     Result. 

28.  32:+ 7  =  19.  37.  3a;+4  =  2a;+5. 

29.  2a:-5  =  ll.  38.    62;+ 8  =  5  a:+ H. 

30.  32:-5  =  0.  »              39.  82:-3-5a:-ll  =  0. 

31.  5a;-8  =  -13.  40.   62;- 12  =  3^;- 3. 

32.  3a;-7  =  -9.  41.  42^-5=28+^. 

33.  3-22^=6.  42.  17-32^=45-102;. 

34.  5  4- 3  2;  =  5.  43.  52^  +  1- 42;=— 2. 

35.  4— 32;  =  4.  44.  32^+1  =  62:— 1. 

36.  -18  2:4- 5  = -10.  45.   -5  2: -10  =  -10 +  3  2:. 

F.  H.  8.  FIRST  YEAR  ALG.  —  6 


66  TBE  SIMPLE  EQUATIOn 

46.  Solve  5a;-[2a;-(a^-2a;-l)]  =  5. 

5  a;  -  [2  x  -  (x  -  2  a:  +  1)]  =  5, 

5a;-[2a;-a:  +  2a;-l]  =5, 
bx-2x-\-x-2x  +  l  =  b, 

bx-2x-\-x-2x  =  b-l, 
2x  =  i, 

X  =  2.    Result. 

47.  4:X-(2x-\-l)=15. 

48.  (5a:-3)-18  =  3a:-(2-a;). 

49.  12-(3a:+7)  =  3-(2:r-5). 

50.  4 a:-  [2ri:-a;  +  l]  =  0. 

51.  4a;-5(2;  +  l)-[2rz;-(a;  +  l)]=l. 

52.  -  i2rr-(a;  +  l)]=13-(2:r-l)-3a;. 

53.  Solve  (ix-\-S')(2x-5)  =  2(x-2y~2(x+l) 

2  x^  +  X  -  15  =  2(x^  -^ X -\-  i)-(2  X  +  2) 
•^^^2  +  a:  _  1 5  =>2^2  _  8  a;  +  8  -  2  x  -  2 , 
+  a:  +  8a:  +  2a:  =  8-2  +  15, 
11  a:  =  21, 

X  =  i\.     Result. 

54.  Cx-l)(x-\-2)  =  (x-S)(x-2y 

55.  (2rr-5)(2^-3)  =  (a;-l)(2:  +  l)+a:2. 

56.  (x-5y-^S(x-h2y=^(a^-l)-S, 

57.  2(2;  -  1)2  -  3(0;  -  2)2  =  6  -  (a;  -  1)2. 

58.  3[2aj-4(3rir2+l)]  =  lT-4(3a:-l)2. 

59.  -  [2(a:-4)(2:-5)-(rr-3)(2:-4)]  =  (5-a:)(a;-2> 

60.  (a;-2)3-(2;-l)3+(3a;-2)(a;-2)  =  0. 


THE  SOLUTION  OF  PROBLEMS  67 

THE  SOLUTION  OF  PROBLEMS 

61.  A  Problem  is  a  question  to  be  solved. 

In  a  problem  certain  conditions  are  given  in  which  an 
unknown  number  is  involved.  It  is  the  value  of  this  un- 
known number  that  we  seek.  And,  assisted  by  the  state- 
ment of  the  conditions  that  relate  to  that  number,  we 
form  an  equation  and  obtain  its  value. 

LITERAL  SYMBOLS  FOR  UNKNOWN  QUANTITIES 

62.  If  a  tank  holds  a  certain  number  of  gallons,  five 
tanks  of  the  same  kind  will  hold  just  five  times  as  many- 
gallons.     That  is, 

If  1  tank  holds  10  gallons,  5  tanks  hold  (5  x  10)  gallons  =  50  gallons. 

If  1  tank  holds  20  gallons,  4  tanks  hold  (4  x  20)  gallons  =  80  gallons. 

If  1  tank  holds  x  gallons,  5  tanks  hold  (5  •  x)  gallons  =  5  a:  gallons. 

If  1  tank  holds  y  gallons,  m  tanks  hold  (m  •  y)  gallons  =  my  gallons. 

Also : 

If  A  has  $100  and  B  has  $50  more  than  A, 

B  has  (100  +  50)  dollars. 
If  A  has  X  dollars  and  B  has  f  50  more  than  A, 

B  has  (a;  +  50)  dollars. 
If  A  has  X  dollars  and  B  has  y  dollars  more  than  A, 

B  has  (x  +  y)  dollars. 

Exercise  15 
Oral 

1.  If  X  denotes  a  certain  number,  write  an  expression 
for  10  more  than  x. 

2.  If  y  denotes  a  certain  number,  write  an  expression 
for  12  less  than  y. 


68  THE  SIMPLE  EQUATION 

3.  If  X  denotes  the  number  of  bushels  of  apples  that  a 
certain  barrel  will  hold,  how  many  bushels  of  apples  will 
there  be  in  8  similar  barrels  ? 

4.  John  solved  x  examples  and  William  solved  y  ex- 
amples.    How  many  did  both  together  solve  ? 

5.  A  boy  had  m  marbles  and  lost  n  of  them.  How 
many  had  he  left  ? 

6.  A  boy  earned  x  cents  and  found  twice  as  many. 
How  many  cents  had  he  in  all  ? 

7.  A  boy  solved  a  examples  and  his  sister  solved  five 
more  than  that  number.     How  many  did  she  solve  ? 

8.  John  piled  x  cords  of  wood  and  Charles  piled  3 
cords  less  than  John.  How  many  cords  did  both  to- 
gether pile  ? 

9.  John  solved  x  examples  and  William  solved  the  same 
number  less  seven.  How  many  examples  did  William 
solve  ? 

10.  Three  men  together  pay  a  bill.  B  pays  twice  as 
much  as  A.  C  pays  three  times  as  much  as  A.  How 
much  does  each  pay  if  A  pays  x  dollars? 

11.  A  library  chair  cost  y  dollars,  a  bookcase  x  dollars, 
and  a  table  as  much  as  the  chair  and  bookcase  together. 
How  much  did  all  together  cost  ? 

12.  A  line  is  10  inches  long  and  x  inches  are  added  to 
it.     What  is  the  increased  length  ? 

13.  From  a  line  x  inches  long  y  inches  are  cut  off. 
How  much  of  the  line  remains? 


THE  SOLUTION  OF  PROBLEMS  69 

14.  Three  lines  of  lengths,  a,  5,  and  c,  respectively,  are 
placed  in  one  straight  line ;  and  c?  inches  are  cut  from 
the  whole.     What  is  the  length  left  ? 

15.  The  sum  of  two  numbers  is  40  and  the  smaller, 
number  is  x.     What  is  the  greater  number? 

16.  The  sum  of  three  numbers  is  50.  One  of  them  is 
X,  another  20.  What  is  the  expression  for  the  third 
number  ? 

17.  The  smaller  of  two  numbers  is  i/  and  the  larger  is 
X.     What  is  the  difference  between  them  ? 

18.  Write  three  consecutive  numbers  if  the  least  of  the 
three  is  x. 

19.  Write  four  consecutive  numbers  if  the  least  of  the 
four  is  m. 

20.  Write  five  consecutive  numbers,  the  greatest  being  a. 

21.  Write  five  consecutive  numbers,  the  middle  one  of 
the  five  being  n, 

22.  What  is  the  next  odd  number  above  m  when  m  it- 
self is  even  ? 

23.  What  is  the  next  odd  number  above  m  when  m 
is  odd  ? 

24.  Write  the  three  consecutive  odd  numbers  below  c. 

25.  Write  the  five  consecutive  even  numbers  above  d, 
d  being  even. 

26.  What  is  the  sum  of  the  three  consecutive  even 
numbers  next  below  k^  k  being  odd  ? 

27.  Write  five  consecutive  odd  numbers  so  that  the 
middle  one  shall  be  m. 


70  THE  SIMPLE  EQUATION 

28.  Write  the  product  of  three  consecutive  even  num- 
bers, the  middle  one  being  x. 

29.  If  a  man  is  x  years  old  now,  how  old  will  he  be  in 
10  .years  ? 

30.  If  a  man  is  y  years  old  now,  how  ol^  was  he  twelve 
years  ago  ? 

31.  A  man  is  twice  as  old  as  his  son  whose  age  is  x 
years.     What  is  the  sum  of  their  ages  ? 

32.  A  man  is  three  times  as  old  as  his  son  and  six 
times  as  old  as  his  daughter.  What  is  the  sum  of  their 
ages  if  the  daughter's  age  is  x  years  ? 

33.  How  old  was  a  man  m  years  ago  if  his  age  at  pres- 
ent is  a  years  ? 

34.  A  man  earns  x  dollars  a  year  for  a  period  of  y  years. 
In  that  time  he  has  spent  z  dollars.  Write  the  expression 
for  his  savings. 

35.  X  and  y  are  two  numbers  and  x  is  the  larger  of  the 
two.  Express  the  fact  that  four  times  the  difference  of 
the  two  is  equal  to  their  sum. 

36.  William  has  a  wallet  containing  x  dimes.  How 
many  cents  has  he  ? 

37.  Express  the  condition  that  h  dimes  shall  equal  m 
nickels. 

38.  A  has  X  dimes  and  y  nickels  in  his  pocket  and  he 
spends  10  cents.  Write  the  expression  for  the  amount  he 
has  left. 

39.  A  man  travels  a  miles  an  hour  for  a  period  of  m 
hours.     How  many  miles  does  he  travel  in  all  ? 

40.  A  man  travels  m  miles  in  a  period  of  x  hours. 
What  is  his  rate  of  traveling  in  miles  per  hour  ? 


THE  SOLUTION  OF  WRITTEN  PROBLEMS  71 

41.  A  boy  rides  x  miles  in  a  boat.  Then  he  walks 
2  miles.  Then  by  train  he  goes  twice  as  far  as  he  has 
already  traveled.  Write  the  expression  for  the  total 
number  of  miles  in  his  journey. 

42.  Write  the  expression  for  the  statement  that  the 
square  of  the  sum  of  a  and  6  is  3  less  than  the  square  of 
the  difference  between  3  m  and  x, 

SUGGESTIONS  FOR  THE  SOLUTION  OF  WRITTEN^  PROBLEMS 

From  the  foregoing  oral  exercise  it  will  be  clear  to  the 
student  that  no  rule  can  be  given  that  will  cover  all  cases 
of  problems.  The  following  suggestions,  however,  will 
give  a  general  outline  as  to  the  method  by  which  we 
reach  a  solution. 

63.   In  solving  a  problem : 

1.  Study  the  problem  to  find  that  number  whose  value  is 
required. 

2.  Represent  this  unknown  number  or  quantity  by  x. 

3.  The  problem  will  state  certain  existing  conditions  or 
relations.     Express  those  conditions  in  terms  of  x. 

4.  Some  statement  in  the  problem  will  furnish  a  verbal 
equation.  Express  this  equation  algebraically  by  the  aid  of 
your  own  written  statements. 

To  aid  the  beginner  a  classification  of  four  common 
types  of  problems  has  been  made,  and  solutions  for  each 
group  are  given  wherever  new  elements  are  introduced. 
After  these  four  groups  a  collection  of  miscellaneous,  un- 
classified problems  gives  abundant  opportunity  for  the 
application  of  principles  already  learned. 


72  THE  SIMPLE  EQUATION 

The  four  groups  are  as  follows : 

1.  Problems  involving  one  number. 

2.  Problems  involving  two  or  more  numbers. 

3.  Problems  involving  the  element  of  time. 

4.  Problems  involving  the  element  of  value. 

Exercise  16 

Problems  involving  One  Number 

* 

1.  Four  times  a  certain  number  is  36.     Find  the  number. 
We  will  represent  the  unknown  number  by  x,  and  we  write : 
Let  X  =  the  required  number. 
Then,  from  the  given  condition, 

4  a;  =  four  times  that  number. 

But  the  problem  states  that 

36  =  four  times  that  number. 

Hence,  from  our  assumed  condition  and  fiom  the  given  condition, 
we  have  4  x  and  36  representing  the  same  quantity, 
From  which,  our  equation, 

4  a:  =  36. 

x=9.  I 

Therefore,  the  required  number  is  9. 

2.  A  certain  number  increased  by  10  equals  25.     Find 
the  number. 

3.  If  5  is  taken  from  a  certain  number,  the  remainder 
is  13.     Find  the  number. 

4.  Three  times  a  certain  number  is  diminished  by  7 
and  the  remainder  is  17.     What  is  the  number  ? 


THE  SOLUTION  OF  WRIT!  J]N  PROBLEMS  73 

5.  John  has  three  times  as  i:  ,ny  books  as  William. 
Together  they  have  20  books  -low  many  books  has 
William  ?     How  many  has  Jo \m  I 

6.  If  three  times  a  certain  number  is  added  to  five 
times  the  same  number,  the  sum  is  72.     Find  the  number. 

7.  Four  times  a  certain  niimbeY  is  subtracted  from 
seven  times  the  same  number  und  the  remainder  is  36. 
Find  the  number. 

8.  By  adding  12  to  a  cei  tain  number  I  get  a  result 
five  times  the  original  numbji*.  What  was  the  original 
number  ? 

9.  I  double  a  certain  number  and  subtract  9  from  my 
result.  My  remainder  is  3  more  than  my  original  number. 
What  was  the  number  ? 

Let    .  X  =  the  required  number. 

2  X  =  double  the  number. 
2  a:  —  9  =  9  less  than  double  the  number. 
Also,  X  -\-  S  =  S  more  than  the  original  number. 

From  the  conditions  of  the  problem  the  last  two  expressions  of  the 
statement  are  equal,  hence  our  equation  : 
2x-9  =  x-[-3. 
2x-x  =  S  +  9. 

a:  =  12,  the  required  number. 

10.  If  5  is  added  to  a  certain  number,  the  sum  is  equal 
to  3  less  than  three  times  the  original  number.  Find  the 
number. 

11.  Three  is  subtracted  from  5  times  a  certain  number, 
and  the  remainder  is  5  more  than  the  original  number. 
What  was  the  number  ? 

12.  Find  that  number  which,  if  doubled,  exceeds  60  by 
as  much  as  the  number  itself  is  less  than  60. 


74  THE  SIMPLE  EQUATION 

Exercise  17 
Problems  involving  Two  or  More  Numbers 

1.  The  sum  of  two  numbers  is  21,  and  the  greater  num- 
ber is  twice  the  smaller  one.     What  are  the  numbers  ? 

Let  X  =  the  smaller  number. 

Then  2x  =  the  larger  number. 

{x  -}-  2  x)  or  3  X  =  their  sum. 
From  the  problem,  21  =  their  sum. 

Hence,  3  a;  =  21. 

X  =  7,  the  smaller  number. 
2  X  =  14,  the  larger  number. 

2.  There  are  two  numbers,  one  of  which  is  7  more  than 
the  other  and  their  sum  is  31.     Find  them. 

Let  X  =  the  smaller  number. 

Then  a;  +  7  =  the  larger  number. 

2  a;  +  7  =  the  sum  of  the  numbers. 
From  the  problem,  31  =  the  sum  of  the  numbers. 

Hence,  2  a:  +  7  =  31. 

2a;  =  31 -7. 
2a:  =  24. 
X  =  12,  the  smaller  number, 
a;  +  7  =  12  +  7  =  19,  the  larger  number. 

3.  There  are  two  numbers,  one  of  which  is  four  times 
the  other.  Their  sum  is  12  more  than  twice  the  smaller 
number.     Find  the  numbers. 

4.  There  are  two  numbers  whose  difference  is  24,  and 
the  greater  number  is  3  more  than  twice  the  smaller  num- 
ber*    What  are  the  numbers  ? 


THE  SOLUTION  OF  WRITTEN  PROBLEMS  75 

5.  The  sum  of  three  numbers  is  45.  The  second  num- 
ber is  twice  the  first  number,  and  the  third  is  equal  to 
twice  the  sum  of  the  first  and  second.  Find  the  three 
numbers. 

6.  The  sum  of  three  numbers  is  25.  The  third  num- 
ber is  twice  the  first  number,  and  the  second  is  5  Ifess  than 
the  third.     Find  the  numbers. 

7.  Find  three  consecutive  numbers  whose  sum  is  45. 
Let  X  =  the  smallest  number. 

Then  a:  +  1  =  the  next  larger  number. 

X  -\-2  =  the  largest  number. 
3  a:  +  3  =  their  sum. 
Hence,  3  a:  +  3  =  45. 

3a;  =  45 -3. 
3a:  =  42. 
X  =  14,  the  smallest  number, 
a;  +  1  =  15,  the  next  larger  number, 
a;  +  2  =  16,  the  largest  number. 

8.  Find  three  consecutive  odd  numbers  whose  sum 
shall  be  21. 

(Let  x,x  +  2,x  +  4:,  represent  the  numbers,  and  state  like  Problem  7.) 

9.  Find  the  five  consecutive  even  numbers  whose  sum 
is  equal  to  seven  times  the  least  number. 

10.  Find  four  consecutive  odd  numbers  such  that  the 
sum  of  the  three  smallest  shall  be  23  less  than  four  times 
the  greatest  one. 


76  THE  SIMPLE  EQUATION 

11.  Divide  19  into  two  parts  such  that  the  smaller  part 
plus  twice  the  larger  part  shall  be  29. 

Let  X  =  the  smaller  part. 

Then  19  —  x  =  the  larger  part. 

2(19  —  x)  =  twice  the  larger  part. 
Hence,  from  the  given  conditions, 

x  +  2(19  -x)  =  29. 
Solving,  X  =  9,  the  smaller  part. 

And  19  -  a:  =  19  —  9  =  10,  the  larger  part. 

12.  Divide  72  into  parts  such  that  3  times  the  larger 
part  added  to  5  times  the  smaller  part  shall  be  274. 

13.  Divide  100  into  two  parts  such  that  twice  the  larger 
part  shall  be  4  more  than  five  times  the  smaller  part. 

14.  Divide  59  into  two  parts  such  that  twice  the  smaller 

part  shall  be  2  less  than  twice  the  larger  part. 

15.  Divide  28  into  parts  so  that  14  less  than  three  times 
the  larger  part  shall  be  equal  to  three  times  the  small  part 
increased  by  4. 

Exercise  18 

Problems  involving  the  Element  of  Time 

1.  A  boy  is  5  years  older  than  his  sister.  In  4  years 
the  sum  of  their  ages  will  be  21  years.  J'ind  the  present 
age  of  each. 

Let  X  =  the  present  age  of  the  sister. 

Then  x  +  5  9=  the  present  age  of  the  boy. 

a;  +  4  =  the  sister's  age  4  years  from  now. 
X  -\-  9  =  the  boy's  age  4  years  from  now. 
From  which       2  a:  +  13  =  the  sum  of  their  ages  4  years  from  now. 


THE  SOLUTION  OF  WRITTEN  PROBLEMS 


11 


Hence,  our  equation, 

2  a; +  13  =  21. 
Solving,  X  =  4:,  the  sister's  age  now. 

a;  +  5  =  9,  the  boy's  age  now. 

2.  A  man  is  twice  as  old  as  his  son  and  the  sum  of 
their  ages  after  10  more  years  will  be  92  years.  Find 
the  present  age  of  each. 

3.  Five  years  ago  the  sum  of  the  ages  of  A  and  B 
was  40  years.  B  is  now  four  times  as  old  as  A.  What 
is  the  present  age  of  each  ? 

4.  In  seven  years  the  sum  of  A's  age  and  B's  age  will 
be  8  years  less  than  five  times  A's  present  age.  At  the 
present  time  A  is  three  times  as  old  as  B.  Find  the  age 
of  each  after  seven  years. 

5.  A  man  is  twice  as  old  as  his  brother.  Five  years 
ago  he  was  three  times  as  old.  Find  the  present  age 
of  each. 

Let  X  =  the  brother's  age  now. 

2  X  =  the  man's  age  now. 
X  —  6  =  the  brother's  age  five  years  ago. 
2x  —  6  =  the  man's  age  five  years  ago. 
Hence,  d(x  —  5)  =  2  a;  —  5. 

Solving,  X  =  10,  the  brother's  age  now. 

2  a:  =  20,  the  man's  age  now. 

6.  In  seven  years  a  man  will  be  twice  as  old  as  his 
brother.  The  sum  of  their  present  ages  is  31  years.  Find 
the  present  age  of  each. 


78  THE  SIMPLE  EQUATION 

7.  A  man  is  five  times  as  old  as  his  sister  but  in  four 
years  he  will  be  only  three  times  as  old.  What  is  the 
present  age  of  each  ? 

8.  A  man  50  years  of  age  has  a  son  15  years  old.  In 
how  many  years  will  the  father  be  twice  as  old  as  the  son  ? 

9.  One  boy  is  16  years  old  and  another  boy  is  8  years 
old.  How^  many  years  ago  was  the  oldest  boy  three  times 
as  old  as  the  youngest  ?  I 

10.  The  sum  of  the  present  ages  of  a  man  and  his  son 
is  52  years.  In  two  years  the  man  will  be  three  times  as 
old  as  the  son.  What  will  be  the  age  of  each  when  the 
sum  of  their  ages  is  100  years  ? 

Exercise  19 
Problems  involving  the  Element  of  Value  ^ 

1.  Divide  1100  among  A,  B,  and  C,  so  that  B  shall 
have  twice  as  much  as  A,  and  C  $  10  more  than  what  A 
and  B  together  receive. 

Let  X  =  the  number  of  dollars  A  receives. 

Then  2x  =  the  number  of  dollars  B  receives. 

3  a;  +  10  =  the  number  of  dollars  C  receives. 
Hence,  6  a;  +  10  =  the  total  received  by  aU  three. 

6a: +  10  =  100. 
Solving,  a;  =  15,  the  number  of  dollars  A  receives. 

2  a:  =  2  •  15  =  30,  the  number  of  dollars  B  receives. 
3  a:  +  10  =  3  •  15  +  10  =  55,  the  number  of  dollars  C  receives. 

2.  A  has  $  3  less  than  B.  C  has  as  many  dollars  as  A 
and  B  together.  All  three  have  1 26.  How  many  dollars 
has  each  ? 


THE  SOLUTIOJ^  OF   WRITTEN  PROBLEMS  79 

3.  A  certain  number  of  yards  of  cloth  cost  $2  per 
yard,  and  the  same  number  of  yards  of  silk  cost  %  5  per 
yard.  The  total  cost  of  both  lots  was  %  70.  How  many 
yards  of  cloth  are  there  in  each  lot  ? 

Let  X  =  the  number  of  yards  in  each  lot. 

Then  2  x  =  the  value  of  the  cloth  in  dollars. 

5  X  =  the  value  of  the  silk  in  dollars. 
Hence,  7  x  =  the  total  value  of  both  lots  in  dollars. 

From  which  we  write  our  equation, 

7  a;  =  70. 
X  =  10,  the  number  of  yards  in  each  lot. 

4.  12  men  receive  f  31  for  a  day's  work.  A  part  of 
the  men  work  at  the  rate  of  f  2  per  day,  and  the  other 
part  receive  $  3  each  per  day.  How  many  men  worked  at 
each  rate  ? 

Let  X  =  the  number  of  men  working  at  $  2  per  day. 

Then        12  —  x  =  the  number  of  men  working  at  f  3  per  day. 
2x  =  the  total  dollars  paid  to  the  first  lot. 
3(12  —  x)  =  the  total  dollars  paid  to  the  second  lot. 
Hence,  the  equation, 
2  a; +  3(12 -a:)  =  3L 

X  =  5,  the  number  of  men  working  at  $  2  per  day. 
12  —  a:  =  12  —  5  =  7,  the  number  of  men  working  at  $  3  per  day. 

5.  A  man  pays  a  bill  of  1 49  with  five-dollar  and  two- 
dollar  bills,  and  uses  the  same  number  of  each  kind. 
How  many  bills  of  each  kind  were  used  ? 

6.  A  boy  has  $  14  in  two-dollar  bills  and  half-dollars. 
He  has  three  times  as  many  coins  as  he  has  bills.  How 
many  has  he  of  each  ? 


80  THE  SIMPLE  EQUATION 

7.  Eight  oranges  cost  a  fruit  dealer  11.90.  A  portion 
of  the  lot  cost  him  20  cents  per  dozen,  and  the  remainder 
cost  him  30  cents  per  dozen.  How  many  dozen  were  there 
of  each  kind? 

8.  A  man  bought  90  postage  stamps,  the  lot  being 
made  up  of  the  five-cent  and  two-cent  denominations. 
Twice  the  value  of  the  two-cent  stamps  was  the  same  as 
the  value  of  the  five-cent  stamps.  How  many  of  each 
kind  were  purchased,  and  what  was  the  total  amount  paid 
for  them? 

Exercise  20 

Miscellaneous  Problems 

1.  Find  two  numbers  whose  sum  is  80  and  whose  dif- 
ference is  10. 

2.  A  and  B  together  have  190,  but  A  has  110  more 
than  B.     How  many  dollars  has  each  ? 

3.  A  certain  number  when  multiplied  by  4  exceeds  15 
by  as  much  as  the  original  number  is  less  than  15.  What 
is  the  number  ? 

4.  One  of  two  numbers  is  4  more  than  the  other,  and 
the  difference  of  their  squares  is  72.  What  are  the 
numbers  ? 

5.  The  sum  of  three  consecutive  numbers  is  26  more 
than  the  second  number.     What  are  the  three  numbers  ? 

6.  The  difference  between  the  ages  of  a  father  and  son 
is  36  years,  and  the  father  is  four  times  as  old  as  the  son. 
Find  the  age  of  each. 

7.  There  are  two  numbers  whose  sum  is  30,  and  whose 
difference  increased  by  6  equals  the  smaller  number. 
What  are  the  numbers? 


THE  SOLUTION  OF   WRITTEN  PROBLEMS  81 

8.  Divide  70  into  two  parts  such  that  twice  the  larger 
part  shall  equal  three  times  the  smaller  part. 

9.  A  man  divides  $700  among  his  four  sons.  Each 
gets  1 50  more  than  the  next  oldest.  How  much  does  each 
get? 

10.  Divide  23  into  two  parts  such  that  1  less  than  four 
times  the  smaller  part  shall  equal  1  more  than  twice  the 
greater  part. 

11.  Ten  times  a  certain  number  is  as  much  above  13  as 
19  is  above  6  times  the  number.     What  is  the  number  ? 

12.  The  difference  between  the  squares  of  two  consecu- 
tive even  numbers  is  28.     Find  the  numbers. 

13.  The  sum  of  two  numbers  is  13  and  the  difference 
between  their  squares  is  13.     What  are  the  numbers  ? 

14.  In  a  family  of  four  children  the  oldest  is  twice  the  age 
of  the  youngest,  and  the  sum  of  the  ages  of  the  oldest  and 
youngest  equals  the  sum  of  the  ages  of  the  other  two. 
All  together  their  ages  amount  to  54  years.  Find  the  age 
of  each  of  the  four,  the  third  child  being  15  years  old. 

15.  One  number  is  three  times  another.  Subtract  the 
smaller  number  from  15  and  the  larger  from  23,  and  the 
remainders  are  equal.     Find  the  numbers. 

16.  At  a  certain  election  a  total  vote  of  1248  was  cast. 
The  successful  candidate  received  a  majority  of  64.  How 
many  votes  were  cast  for  each  candidate  ? 

17.  A  man  has  four  hours  of  time  at  his  disposal  and  he 
walks  out  into  the  country  at  a  rate  of  4  miles  per  hour. 
How  many  miles  can  he  walk  so  that,  by  returning  on  a 
trolley  car  at  the  rate  of  fifteen  miles  per  hour,  he  will 
return  at  the  end  of  his  time  ? 

F.  H.  S.  FIRST  YEAR  ALG. 6 


82  THE  SIMPLE  EQUATION 

18.  A  walks  over  a  certain  road  at  the  rate  of  4  miles  an 
hour.  Three  hours  after  he  left  his  home,  B  starts  after 
him  at  a  rate  of  5  miles  per  hour.  How  many  miles  will 
A  have  gone  when  B  overtakes  him  ? 

19.  A  and  B  are  48  miles  apart  and  start  at  the  same 
time  to  travel  toward  each  other.  A  goes  at  a  rate  of  3 
miles  an  hour  and  B  goes  at  a  rate  of  5  miles  an  hour. 
How  many  hours  will  pass  before  they  meet,  and  how  far 
will  each  have  traveled  ? 

20.  A  man  paid  $178  for  a  horse  and  a  harness.  The 
cost  of  the  horse  was  f  10  more  than  the  cost  of  five  simi- 
lar harnesses.  What  was  the  cost  of  the  horse  and  the 
harness  ? 

21.  A  man  walked  the  first  10  miles  of  a  journey,  then 
rode  a  certain  distance  in  a  train,  and  finally  traveled  in 
an  automobile  twice  as  far  as  he  had  already  come.  He 
traveled  45  miles  in  all.  How  many  miles  did  he  go  by 
train  and  in  the  automobile  ? 

22.  Ten  yards  of  cloth  and  5  yards  of  silk  cost  in  all 
$60.  The  cost  of  the  silk  was  twice  the  cost  of  the  cloth. 
What  was  the  cost  of  each  per  yard  ? 

23.  How  can  you  pay  a  bill  of  $3.50  so  that  you  will 
use  the  same  number  each  of  dimes  and  quarter  dollars, 
and  no  other  coins? 

24.  The  sum  of  the  ages  of  a  father  and  son  is  92  years ; 
but  if  the  son's  age  is  doubled,  it  will  be  4  years  more 
than  his  father's  age.      Find  the  age  of  each. 

25.  A  had  $7  more  than  three  times  B's  money.  A 
gave  B  $8  and  now  he  has  only  $1  more  than  B.  How 
much  had  each  at  first  ?     How  much  has  each  now  ? 


CHAPTER  VII 
SUBSTITUTION 

64.  Substitution  is  the  process  of  replacing  literal  factors 
in  algebraic  terms  by  numerical  or  by  other  literal  values. 

Thus :  If  a  =  5  and  6  =  7 : 

(1)  (a  +  6)  =  (5  +  7) 

=  12.     Kesult. 

(2)  (2a-a&)  =  (2.5-5.7) 

=  (10  -  35) 

=  -  25.     Result. 

If  x=:2m,  y  =  dm,  and  z  =  6m: 

(3)  (a:  +  ?/  +  2)  =  (2  m  +  3  m  +  5  m) 

=  10m.     Result. 
(4:)   (x  +  y)(2x-y-\-Sz)  =  (2m  +  dm)(2-2m-Sm  +  S-5m) 
=  (5  m)  (4  m  —  3  m  +  15  m) 
=  (5  m)  (16  m) 
=  80  m2.     Result. 

GENERAL  METHOD 

65.  To  substitute  numerical  or  literal  values  in  a  given 
expression : 

Replace  the  literal  factors  of  the  terms  of  the  given  expres- 
sion hy  their  respective  given  values. 

Perform  all  indicated  operations^  and  simplify  the  result. 

83 


84  SUBSTITUTION 

Occasional  problems  will  permit  the  substitution  of  given 
values  either  before  or  after  indicated  operations  are  per- 
formed ;  but,  in  general,  the  better  plan  is  to  make  substi- 
tution the  first  step,  for  in  practical  applications  this  order 
is  almost  universal. 

I.  Substitution  of  Numerical  Values 
Exercise  21 

Find  the  numerical  value  of  the  following,  when  a  =  4, 
6=3,  c  =  2,  and  c?  =  1. 


1. 

a-{-h  -\-  c. 

2.   a—^h  +  Be. 

a 

+&+c=4+3+2 

a-2&  +  5c  =  4-2(3)+5(2) 

=  9.    Result. 

=  4  _  6  +  10 
=  8.    Result. 

3. 

4  6  _  3  «  4.  2  ^. 

5.    lOa-{-h-CSc-d), 

4. 

1  a-d+^c-h. 

6.   ib-lSc-C2a-^d)'], 

7. 

ah  —  ^hc^b  ad. 

ab-dbc^5ad  =  (4)  (3)  -  3  (3)  (2)  + 5(4)  (1) 

=  12- 

-18  +  20 

=  14. 

Result. 

8.  2ab-Sed-{-2bd. 

9.  Bad-^bd-hScd. 

10.  Sabc  —  b  bed  +  2  acd  —  12  abd, 

11.  ab  —  (be  —  cd}. 

12.  abc  —  \_acd  —  (bcd—abd)'\. 

13.  2ab-(Bcd-\-2a-  ab  +  ed}. 

14.  bc-\-[bd-(iab-cd)]+(iab-bc}l 


GENERAL   METHOD  85 

15.  Simplify  a{a  -\-  e)  —  c(c  —  a)  when  a  =  4  and  c  =  3. 

a(a  -f  c)  -  c{c  -  a)  =  4(4  +  3)  -  3(3-  4) 
=  4(7)-3(-l) 

=  28  +  3 

=  31.     Result. 

16.  Simplify  (a  +  6)2  _  (<^  +  6)  (a  -  5)  -  («  -  5)2  when 
a  =  5  and  5  =  2. 

(a-f6)2-(a  +  />)(a-5)-(a-6)2=  (5  +  2)2- (5  +  2)  (5-2) -(5-2)2 

=  (7)2 -(7)  (3) -(3)2 
=  49-21-9 
=  19.    Result. 

17.  (a +  2)2 -(a -1)2  when  a  =  3. 

18.  10(6f+-4)2-(3a  +  2)2  when  a  =10. 

19.  4(2a;+-?/)2— 8(a:+-y)(a:— ?/)  when  x=2  and  ^  =  0. 

20.  3(m  — 2a;)2— 2(w  +  2a;)(m  — 2a;)+-4a;  when  m  =  5 
and  x=l. 

21.  2(c  +  (^)-S2(6'  +  ^)((?-(?)-3(c-(^)S  when  c=l 
and  c?  =  —  1. 

22.  a(a  +  6)-[6(a-6)2  +  (a+-5)3]  when  «=-3  and 
^.=  -4. 

23.  (3  a+-m)(3  «  -  w)  +  [9  a2  _  j^^  _  ^(2  w-  9a)j] 
when  a  =  0  and  m  =  2. 

24.  5a(a+-l)-(2-a)(3<x  +  5)-2a(a+-l)  when  a  =  0. 

25.  (3  w  -  2)2  +  (m  +  w  +  1)2 -  (2 m -  l)(m  +  1)  when 

7W  =  71  =  —  1. 


86  SUBSTITUTION 

II.   Substitution  of  Literal  Values 

When  the  values  given  for  substitution  are  literal 
quantities,  the  process  is  identical  with  that  of  numerical 
substitution.  The  results,  however,  will  be  in  terms  of  the 
literal  quantities  substituted. 

Exercise  22 

Substitute  the  given  literal  values  and  reduce  the  fol- 
lowing: 

1.  (a  +  xy  —  (a  —  xy^  when  a=2x.  ♦; 

(a  +  xy  -(jci-xy  =  {2x-{-  xy  -(2x-  xy 
=(dxy-(xy 

=  8  a;2.     Result. 

2.  (a  —  3  a;  —  x^^  when  x=  —  a. 

3.  (a  4-  a?)  (a  —  x)^  —  (a  +  3  xy  when  a  =  —  x. 

4.  (x  +  m}(x-\-m  —  7i)  when  m  =  x  and  n=  —  x. 

5.  (2  a  —  "^  c  -\-  x)(x  —  a)  —  acx  when  a  =  x  and  c  =  0. 

6.  5  ac  —  (a  —  3  <?)  (a  +  10  c)  —  c  +  ((?  —  a)^  when  <?  =  0. 

7.  2(mH-a;)  — 2(w  +  a;)(77i  — 2:)  — 3  771a;  when  m=a  and 
a;=  3«. 

8.  (2  a;  +  1)2 -3(a:- 1)2  when  a:  =4  a. 

9.  (3a;2-4a;+l)-  (2  a: +  1)2  whena:=-«5. 

10.  (a  4-  a:) (a  —  a:)  —  (a  —  a;)  +  2  a;  when  x=  —  ^  a. 

11.  (a  +  6+l)2-(a+6)2+2(a  +  5)-l  when  a=-x 
and  5  =  —  3  a;. 

12.  (Qx^-\\xy-10y'^^-Qx(x-2y')  +  l^y'^  when 
x  =  m  and  y  —  —  m. 


APPLICATIONS  OF  SUBSTITUTION  87 

APPLICATIONS  OF  SUBSTITUTION 
I.     To   VERIFY   AN   EQUATION 

66.  To  verify  an  equation  is  to  prove  the  truth  of  the 
result  obtained  for  its  root. 

When  we  verify  an  equation  we  show  that  our  result, 
when  substituted  for  x  in  the  given  equation,  reduces  both 
members  to  the  same  quantity. 

In  such  a  case  a  root  is  said  to  satisfy  an  equation. 

Exercise  23 

1.    Show    that  X  =  1   satisfies    the    equation   3  a;  +  4 

=  2  2;  +  5. 

dx-\-i  =  2x  +  6. 

3(1)  +  4  =  2(1)  +  5. 

3  +  4  =  2  +  5. 

7  =  7. 

2.  Show  that  X  =  S  satisfies  the  equation  3  a;  —  4 
=  2:+- 12. 

3.  Show  that  x=  —  4:  satisfies  the  equation  2{x  +  3) 
=  4  +  (r^^-2). 

4.  Show  that  x=0  satisfies  the  equation  5 (rr  —  2) 
=  3(a;+l)_13. 

5.  Show  that  x=  —  l  satisfies  the  equation  (a;  + 1) 
(a;+2)=<a;+l). 

6.  Is  2  a  root  of  the  equation  2(^3p  4-  2  a;  +  1)  —  (a;  +  2) 
=  2a^  +  6? 

7.  Prove  that  —  4  will  satisfy  the  equation  (a;  +  4)^ 
+  (a:  +  l)2=(a;  +  3)2+2a;(a;+l)-a^. 


88  SUBSTITUTION 

8.  Show  that   x  =  0   satisfies  (4a;  -  l)(a;  +  3)  -  da? 

_(_10:r+3)  +  6  =  0. 

9.  Prove  2 c  a  root  of  the  equation  2cx-\-d  =  4:C^-{-d, 

10.    Show  that  x=l  satisfies  3(<x  +  b}x  —  2 (a  —  b)x  = 
a  +  5b. 

II.   The  Use  of  Formulas 

67.   A  Formula  is  an  algebraic  expression  for  a  general 
principle. 

Thus,  if  the  area  of  a  triangle  is 
represented  by  S,  the  base  by  5, 
and  the  altitude  by  a,  the  formula 

S  =  —■  gives  us  an  expression  for 

the  area  of  the  triangle  in  terms  of  its  base  and  altitude. 
Then  if  we  know  the  values  for  the  base  and  the  altitude, 
we  obtain  the  area  by  the  substitution  of  those  values  and 
the  reduction  of  the  formula. 

We  will  now  make  application  of  this  principle  to  a  few 
formulas  in  common  use. 


(a)   The  Circumference  of  a  Circle 

Let  C  =  the  circumference  of  a  circle. 

R  =  the  radius  of  the  same  circle. 

3.1416  =  the  constant  relation  between  the  circumference 
and  the  diameter  of  a  circle.  (This  constant  is 
usually  represented  in  formulas  by  the  Greek 
letter  ir,  called  "  pi.") 

Then  C  =  2  7ri2  is  the  formula  for  the  circumference  of  a  circle 

whose  radius  is  known. 


APPLICATIONS   OF  SUBSTITUTION  89 

Exercise  24 

1.  Find  the  circumference  of  a  circle  whose  radius  is 
10  feet. 

Given  R  =  10, 

TT  =  3.1416. 
Then  C  =  2  7rR 

=  2  X  3.1416  X  10 

=  62.832  feet.    Result. 

2.  Find  the  circumference  of  a  circle  whose  radius  is 
14  inches. 

3.  Find  the  circumference  of  a  circle  whose  diameter 
is  18  feet.  (The  diameter  of  a  circle  is  equal  to  twice  its 
radius.) 

4.  The  circumference  of  a  circle  is  12.5664  feet.  Find 
the  radius  of  the  circle.  (Substitute  the  known  values 
and  solve  the  formula  for  B.) 

5.  The  circumference  of  a  circle  is  31.416  feet.  Find 
its  diameter. 

6.  In  going  one  mile  a  carriage  wheel  turns  440  times. 
Find  the  radius  of  the  wheel.  " 

(b)   The  Area  of  a  Circle 

Let  S  =  the  area  of  a  circle. 

R  —  the  radius  of  the  same  circle. 

TT  =  the  constant,  3.1416. 

Then  S  =  irR^  is  the  formula  for  the  area  of  a  circle 

whose  radius  is  known. 


90  SUBSTITUTION 

Exercise  25 

1.  Find  the  area  of  a  circle  whose  radius  is  10  feet. 
Given  R  =  10. 

TT  =  3.1416. 
Then  S  =  irR^ 

=  3.1416  X  102 
=  3.1416  X  100 
=  314.16  square  feet.     Result. 

2.  Find  the  area  of  a  circle  whose  radius  is  14  inches. 

3.  Find  the  area  of  a  circle  whose  diameter  is  18  feet.  '! 

4.  The  area  of  a  circle  is  28.2744  square  feet.  Find 
the  radius. 

5.  The  area  of  a  circle  is  12.5664  square  feet.  Find 
the  diameter. 

6.  What  is  the  area  of  a  circular  ring  whose  two  diame- 
ters are  10  and  12  feet  respectively  ? 

7.  How  many  rods  of  fence  will  be  needed  to  inclose  a    ; 
circular  park  whose  area  is  176.715  square  rods  ? 

8.  A  man  has  a  circular  pond  whose  diameter  is  100 
feet.  He  builds  around  it  a  walk  6  feet  wide.  What  is 
the  area  of  the  walk  ? 

(c)   TVie  Sum  of  a  Series  of  Numbers  or  Quantities 

68.  A  Series  is  a  succession  of  numbers  formed  accord- 
ing to  some  fixed  law.     Thus  : 

1,  3,  5,  7,  9,  11,  13,  etc., 

is  a  series  in  which  each  term  is  greater  by  2  than  the 
preceding  term. 


APPLICATIONS  OF  JSUBSTlTUTtOK  91 

A  series  formed  in  this  way  is  called  an  Arithmetical 
series. 

When  certain  elements  of  a  series  of  this  kind  are 
known,  others  can  be  found  by  the  use  of  formulas. 
Thus : 

s... 

Let  a  =  the  first  term  of  a  series. 

.  I  =  the  last  term  of  a  series. 

n  =  the  number  of  terms  in  the  series. 

aS  =  the  sum  of  the  terms  in  the  series. 

Then  S  =  ^(a  -{■  I)  is  the  formula  for  finding  the  sum 

of  an  arithmetical  series  when  the  first 
term,  the  last  term,  and  the  number  of 
terms  are  known. 

Exercise  26 

1.  The  first  term  of  an  arithmetical  series  is  2,  the  last 
term  18,  and  the  number  of  terms  9.  Find  the  sum  of 
the  terms  in  the  series. 

From  the  statement,     a  =  2,  I  =  18,  n  =  9. 

Hence,  5  =  ^(a  +  Z) 

=  |(2  +  18) 
=  90.     Result. 

2.  There  are  10  terms  in  an  arithmetical  series.  The 
first  term  is  5  and  the  last  term  15.  What  is  the  sum  of 
the  terms  in  the  series  ? 

3.  The  sum  of  a  series  of  sixteen  terms  is  136,  and  the 
last  term  of  the  series  is  16.  What  is  the  first  term  of 
the  series  ? 


92  SUBSTITUTION 

4.  A  certain  arithmetical  series  has  a  first  term  12, 
and  a  last  term  18.  The  sum  of  the  terms  is  105.  How 
many  terms  are  there  in  the  series  ? 

5.  The  first  term  of  an  arithmetical  series  of  nine  terms 
is  —  35,  the  last  term  —  3.     Find  the  sum  of  the  terms. 

6.  The  first  term  of  a  series  of  six  terms  is  —  5  x^  and 
the  second  term  is  5  x.     Find  the  sum  of  the  series. 

7.  The  sum  of  the  terms  of  an  arithmetical  series  is 
21  x  +  33.  The  first  of  the  six  terms  in  the  series  is  2  a; 
+  3.     Find  the  last  term. 

8.  The  sum  of  a  succession  of  odd  numbers  that  began 
with  7  is  176.  How  many  numbers  are  there  in  the  series 
if  the  last  number  is  37  ? 

(d)    The  Formula  for  Falling  Bodies 

When  a  body  falls  to  the  earth,  it  gains  a  velocity  of  32 
feet  per  second  for  each  second  of  its  fall.  The  following 
formulas  are  in  common  use  for  experiments  with  falling 
bodies. 

Let  a  =  the  gain  in  velocity  per  second. 

t  =  the  number  of  seconds  during  which  the  body  falls. 
S  =  the  space  through  which  the  body  falls. 
Then,  (1)  S  =  lat"^  is  the  formula  for  finding  the  space  passed 

over  by  a  falling  body  in  a  given  time. 
Again,  let   a  =  the  gain  in  velocity  per  second. 

t  —  the  number  of  seconds  during  which  the  body  falls. 
V  —  the  velocity  of  the  body  at  the  end  of  any  given 
second. 
Then,  (2)    v  =  a<  is  the  formula  for  finding  the  velocity  of  a  fall- 
ing body  at  the  end  of  any  given  second. 
For  a  we  use  32  feet,  or,  in  the  Metric  System,  980  centimeters. 


APPLICATIONS  OF  SUBSTITUTION  93 

Exercise  27 

1.  In  2  seconds  a  ball  fell  from  the  top  of  a  flag  pole 
to  the  ground.     How  many  feet  did  the  ball  fall  ? 

Given  a  =  32  feet, 

t  =  2  seconds. 
Hence,  5  =  -|  at^ 

=  i  X  32  X  4 

=  64  feet.     Result. 

2.  A  ball  dropped  from  the  top  of  a  tower  struck  the 
ground  in  3  seconds.     How  high  was  the  tower  ? 

3.  A  coin  was  dropped  from  the  edge  of  a  cliff  and  it 
struck  the  water  in  6  seconds.     How  far  did  the  coin  fall  ? 

4.  What  is  the  velocity  of  a  ball  at  the  end  of  the  3d 
second  of  its  fall  ? 


Given 

a  =  32  feet, 

<  =  3,  the  number  of  the  second  considered 

Hence, 

V  =  at 

=  32  X  3 

=  96  feet.     Result. 

That  is,  the  ball  is  falling  at  a  rate  of  96  feet  per  second  at  the 
end  of  the  third  second. 

5.  A  stone  fell  from  a  roof  and  fell  for  5  seconds  before 
striking  the  ground.  How  far  did  the  stone  fall  ?  What 
was  its  velocity  at  the  end  of  the  4th  second?  At  the 
end  of  the  5th  second  ? 

6.  The  velocity  of  a  ball  when  it  struck  the  ground 
was  found  to  be  288  feet  per  second.  For  how  many 
seconds  had  the  ball  been  falling  ?  What  was  its  velocity 
at  the  end  of  the  7th  second  ? 


CHAPTER  VIII 

SPECIAL     CASES     IN     MULTIPLICATION    AND 
DIVISION 


MULTIPLICATION 


The  product  of  vsimple  forms  of  binomials  may  often  be 
obtained  without  actual  multiplication.  The  following 
examples  will  show  the  principles  upon  which  the  abbre- 
viated processes  depend. 


The  square  of  the 
sum  of  two  quanti- 
ties 


II. 

The  square  of  the 

difference  of  two 

quantities 


III. 

The  product  of  the 

sum  and  difference 

of  two  quantities 


a  + 
a  + 


a  + 

a  — 


a2  + 


ab 


a2- 


ab 

ab  +  b^ 


a^  +     ab 


a^  +  2ab-hb^ 


a^-2ab  +  b^ 


-     ab-b^ 
-b^ 


From  these  actual  multiplications  we  make  the  follow- 
ing conclusions : 

From  I.  :  *• 

69.  The  square  of  the  sum  of  two  quantities  equals  the 
square  of  the  firsts  plus  twice  the  product  of  the  first  hy  the 
second^  plus  the  square  of  the  second. 

94 


I 


MULTIPLICATION  95 


L    From  II. : 


70.  The  square  of  the  difference  of  two  quantities  equals 
the  square  of  the  first,  minus  twice  the  product  of  the  first 
hy  the  second,  plus  the  square  of  the  second. 

I     From  III.  : 

71.  The  product  of  the  sum  and  difference  of  two  quan- 
tities equals  the  difference  of  their  squares. 

Exercise  28 
Write  by  inspection  the  following  indicated  products : 

1.  {a  +  x)\  4.    («  +  2)2.  7.    (2a+3)2. 

2.  (^x  +  yy.  5.    (a  4- 2^)2.  8.    (3  a +  5)2. 

3.  (2:4-2)2.  6.    (a  +  5c)2.  9.    (4  a; +3)2. 

10.  (m-w)2.        13.    (a -5)2.  16.    (a6-7)2. 

11.  (2; -5)2.         14.    (2  a -3)2.  17.    (^252- 5)2. 

12.  (a- 7)2.         15.    (5  a- 2)2.  18.    (3am-2ma;)2. 

19.  (a  +  6)(a-6).  23.  (4a  +  10)(4a-10). 

20.  {m -\- x) (^m  —  x') .  24.  (5  w  +  7?z)(5?w— 7w). 

21.  (a+5)(a-5).  25.  (2  ^2  + 9)(2  a2- 9). 

22.  (2a  +  3)(2a-3).  26.  (6  a5  +  5)(6  a5- 5). 

27.  (2a5  +  3  5c)2.  31.  (^2x^  +  ll}(2a^-ll). 

28.  (4  a;2?/2  —  7)2.  32.  (ax^  +  xy^Qaoc^  —  xy). 

29.  (3m3-10)(3m3  +  10).  33.  (2^5  +  11)2. 
\     30.  (bxy-\-^xzy.  34.  (5m7-8)2. 


96  SPECIAL   CASES 

IV.     The  Difference  of  Two  Squares  obtained  from 
Trinomials 

Expressions  that  contain  three  terms  may  be  grouped 
in  many  cases  so  as  to  come  under  the  principles  that 
govern  the  binomial  cases  already  considered.  The  pro- 
cess depends  upon  the  grouping  of  two  of  the  three  given 
terms  in  a  parenthesis^  this  parenthesis  being  treated  as  one 
term. 

Three  different  groupings  are  possible,  as  follows : 

I.  II. 

(a  +  5  4-  c)(a  +  h  -c)  (a  +  5  -f-  c)(«  -  ^  +  c) 

III. 

(a  +  h  —  c){a  —  h  -\-  c) 

The  terms  inclosed  in  parentheses  must  always  he  like  in 
sign. 

From  I.      (a  +  &  +  c) (a  +  J  -  c)  =  [(a  +  &)  +  c] [(a  +  i)  -c] 

=  (a  +  hy  -  c2 
=  a2  +  2  a&  +  &2  _  c2.     Result. 

From  II.     {a-^l)  +  c){a-h+c)  =  [(a  +  c)  +  &][(a  +  c)  -  ft] 

=  (a  +  c)2  -  6-2 
=  a2  +  2  ac  +  c2  -  h\     Result. 

From  III.  In  this  case  only  one  term  has  the  same  sign  in  each 
expression,  —  the  a  term.  Hence,  the  last  two  terms  of  each  expression 
must  be  inclosed  in  parentheses.  The  sign  of  the  second  term  of 
each  expression  becomes  the  sign  of  the  parenthesis. 

(a  +  &  -  c)  (a  -  &  +  c)  =  [a  +  (&  -  c)]  [a  -  (6  -  c)] 
=  a^-  (h-cY 
=  a^-(h'^-2bc-i-c'^) 
=  a^-b^-\-2bc-  c\    Result. 


MULTIPLICATION  97 

Exercise  29 

Write  by  inspection  the  following  indicated  products : 

1.  [(a-i-c}-\-x']l(a  +  c)-x'].  9.    (x^-\-x-^lX^  +  ^-'^)' 

2.  [(a+5)  +  2]  [(a+^)-2].       10.    (a;-a  +  5)(a;-a-2>). 

3.  [(a-2)-{-x'][(a-2)-x].       11.    (m-x+lXm-^-x+iy 

4.  l(x^-^l)+x]l(a^  +  l)-x'].    12.    (c-x-\-2Xc-\-x  +  2'), 

5.  [a+(tf+tZ)]  [a-((?+^)].       13.   (a  +  w-l)(a-m4-l). 

6.  [m  +  (a;-^)]  [^-(a;-?/)].     14.    (a4-^-5)(a  — ^  +  5). 

7.  (fl^  +  c  +  a;)C«4-c-:K).  15.    (c2-2-t?)(c2_2  +  <?). 

8.  (a4-5  +  3)(a  +  &-3).  16.    (tf2_^^_2)(c2_c_|.2). 

17.  [(a  +  a:)  +  C^  +  l)][(a  +  a:)- (2/4-1)]. 

18.  [(a  +  ^)-(c?-c?)]  [(a  +  5)+(c-(^)]. 

19.  (a-b  +  c-l)(a-^h  +  c-hl}. 

20.  (a3-a2_^_l)(^3_^^2^_^_l), 

V.  The  Product  of  Binomials  having  a  Common  Literal  Term 

The  actual  multiplication  of  the  binomials  (a  +  7)  and 
(a  +  5)  gives  ^  ^   7 

a   +    o 
a2+    7  a 

+    5  a  +  35 
a2  +  12  a  +  35 

The  terms  of  the  product  are  made  up  as  follows  r 
a"^  =  a  '  a,  the  product  of  the  given  first  terms. 

+  12  a  =  (+  7  +  5)  a,    the  product  of  the  common  literal  term  by  the 
sum  of  the  given  last  terms. 
+  35  =  (+  5)  (+  7),  the  product  of  the  given  last  terms. 

F.  H.  S.  FIRST  TEAR  ALG.  — 7 


98  SPECIAL   CASES 

In  the  same  manner  we  may  obtain  the  product  of  the 
binomials  (x  —  4)  and  (x  —  9). 

x:x  =  x^,  (_4-9)a;  =  -13a:,         (-4)  (-9)  =  +  36. 

Hence,  (x  -  i)  (x  -  9)  =  x^ -13  x -\-  36.     Result. 

Or,  in  a  case  where  the  signs  of  the  second  terms  are  unlike, 
(x -\- 9)  (X  -  S). 

x-x  =  x%  (+  9  -  3)  a;  =  +  6  a:,         (+  9  )  (  -  3)  =  -  27. 

Hence,  (ar  +  9)  (x  -  3)  =  x^  +  6  a:  -  27.     Result. 

And,  again,  the  product  of  (m  —  15)  (m  +  7). 

m'm  =  m%        (-  15  +  7)  wi  =  -  8  w,         (_  15)  (+  7)  =  -  105. 

Hence,  (m  -  15)  (?«  +  7)  =  m^  -  8  m  -  105.     Result. 

From   these    illustrations    we    may   state    the    general 
process. 

72.   The  product  of  two  binomials  having  a  common 
literal  term  is  obtained  as  follows : 

The  first  term  is  the  product  of  the  given  first  terms. 
The  second  term  is  the  product  of  the  common  literal  term 
hy  the  algebraic  sum  of  the  given  second  terms. 

The  third  term  is  the  product  of  the  given  second  terms. 

Exercise  30 
Write  by  inspection  the  following  products : 

1.  (a +  2) (a +  3).  6.  (a  +  12)(a  +  l). 

2.  (a  +  5)(«  +  3).  7.  (a +11)  (a +10). 

3.  (c+6)(c?+7).  8.  (a:  +  7)(rr  +  20). 

4.  (w  +  5)(m  +  9).  9.  (re +  16)  (a; +  9). 

5.  (a: +  8)  (a; +10).  10.  (2:  + 14)  (2;  + 15). 


MUL  TIP  Lie  A  TION  99 

11.  (a -4) (a -5).  16.  (a-12)(a-10). 

12.  (a;-3)(rr-4).  17.  (a5  -  7)  (a6- 30). 

13.  (a; -8)  (37-1).  18.  (rr^  -  15)  (a;?/ -  3). 

14.  (m-7)(w-4).  19.  (c2_i0)(c2_20). 

15.  (3/ -10)  (2/ -3).  20.  (c2c22_7)(^^_ll). 

21.  (2; +  5)  (a: -3).  28.   (2:2  +  6)(a;2_ii). 

22.  (a;+3)(2:-5).  29.    (a2_  5)  (^2  +  13). 

23.  (6J_4)(a  +  7).  30.    (w3+7)(^3_4). 

24.  (a +  4)  (a -7).  31.  (a52_9)(^52  +  20). 

25.  (w-9)(7yi  +  10).  32.  (a;2?^2  +  io)(^^2_i9), 

26.  (2; +  5)  (2; -9).  33.  ((?3(^3_i5)(^^  +  2). 

27.  ((?-12)(c+ll).  34.  (ax^-S0x)(a2^-{-2x), 

VI.    The  Product  of  Any  Two  Binomials 

An  application  of  the  principle  already  used  in  the  fore- 
going exercise  enables  us  to  obtain  the  product  of  any  two 
binomials. 

By  actual  multiplication : 

2a  +    5 
3a  +    4 


+    8  g  +  20 
6  a2  +  23  a  +  20 

The  two  multiplications  that  result  in  the  terms  -f- 15  a 
and  +S  a  are  called  cross-products. 

In  multiplication  by  inspection  the  student  will  be 
helped  if  he  will  imagine  that  the  terms  entering  into  the 


100  SPECIAL   CASES 

cross-products  are  connected  as  indicated  below.      Then 
in  the  example  above  we  would  have 


(2  a  +  ^)(3  a  +  4) 

From  which  form  we  immediately  write  the  products 
that  together  make  the  middle  term. 

Thus,       (+  2  a)  (+  4)  +  (+  3  a)  (+  5)  =  +  8  a  +  15  a  =  +  23  a. 

Similarly : 

In  (4  a  -  7)  (3  a  -  5)  the  middle  term  is 
(  +  4a)(-5)  +  (  +  3a)(-7)  =  (-20a)+(-21a)=-20a-21a=:-41a. 

In  (2  a  +  5)  (3  a  —  4)  the  middle  term  is 
(  +  2a)(-4)  +  (  +  3a)(  +  5)  =  (-8a)  +  (  +  15a)  =  -8a  +  15a=+7a. 

In  (3  a -2)  (9  a +  4)  the  middle  term  is 
(+3a)(  +  4)  +  (  +  9a)(-2)  =  (  +  12a)  +  (-18a)  =  +12a-18a=-6a. 

From  these  illustrations  we  may  state  the  general  process. 

73.   The  product  of  any  two  binomials  is  obtained  as 
follows : 

The  first  term  is  the  product  of  the  given  first  terms. 
The  second  term  is  the  algebraic  sum  of  the  cross-products 
of  the  given  terms. 

The  third  term  is  the  product  of  the  given  second  terms. 

Exercise  31 

Write  by  inspection  the  following  products : 

1.  (2a-f-3)(a  +  l).  5.   (4a;4-l)(2?:+3). 

2.  (3a:+2)(2rr  +  l).  6.  (3  m4- 1)(2  tw  +  5). 

3.  (2 a:  +  3) (a;  +  3).  7.   (5  a: +  2)  (3  a; -hi). 

4.  (2a  +  l)(3a  +  l).  8.   (7  a:  + l)(a;+ 9). 


MULTIPLICATION  101 

9.  (2a-7)(3«-4).  13.  (5m-7)(8m-5). 

10.  (3a-l)(a-6).  14.  (10  a- 17)  (8  a  -  3). 

11.  (4c-l)(3(?-5).  15.  (7m-5y)(8w-3«^). 

12.  (3^-4)(2?/-3).  16.  (2a-56)(7a-26). 

17.  (4  a  -f-  7  m)  (5  a  -  9  m).  21.  (11  «+ 10  5)  (9  a  -  7  5). 

18.  (7rc-4«/)(8a;  +  9?/).  22.  (7  rr- 13  2/)  (5  a; +  10?/). 

19.  («6-3  2/)(5a5  +  2^).  23.  (a5tf-9)(7a5c  +  ll). 

20.  (a;2/+7  2)(4a:^-2).  24.  (12  a- 5) (5  a  +  12). 

VII.  The  Square  of  Any  Polynomial 

By  actual  multiplication : 
(a  +  6  +  c)2  =  a2  +  &2  +  c2  +  2  a6  +  2  ac  4-  2  &c. 

(a  +  &  +  c  +  c?)2  =  a2  +  &2  +  c2  +  ^2^.  2  aft  +  9  ac  +  2  ac?  +  2  5c  +  2  &rf+2  erf. 
(a-6-c-rf)2  =  a2  +  62  +  c2  +  d2_2a&-2ac-2arf  +  2&c+2M+2crf. 

It  will  be  noted  that 

(1)  All  the  squares  in  the  product  are  positive. 

(2)  The  other  terms  are  positive  or  negative  according 
as  the  signs  of  the  factors  composing  them  are  like  or 
unlike  in  sign. 

(3)  The  coefficient  of  each  product  of  dissimilar  terms 
is  2. 

Applying  these  principles  we  have 

(a-2&  +  3c)2  =  (a)2+  (_2&)2+  (3  c)2  +  2(a)(- 2  &)  +2(a)(3c) 
+  2(-2  6)(3c) 

=  a2  +  4  62  +  9  c2  _  4  a&  +6  ac  -  12  &c. 


102  SPECIAL   CASm 

Hence,  the  process  of  writing  the  square  of  any  poly- 
nomial is  as  follows : 

74.  The  square  of  any  polynomial  is  the  sum  of  the 
squares  of  the  several  terms  together  with  twice  the  product 
of  each  term  into  each  of  the  terms  that  follow  it. 

Exercise  32 

Write  by  inspection  the  following  products : 

1.  (a  +  5  +  a:)2.  9.    Qa-{-h^-m-\-ny, 

2.  (m  +  n-{-xy.  10.    (x  +  a  +  m  +  yy. 

3.  (x  +  y  +  zy,  11.    (^a-h^-c-df, 

4.  (^a  +  h~xy,  12.    (a- 2  w  + 3  a;- 2)2. 

5.  Qm-n-py  13.    (2a-3  6  +  4(?-5c?)2. 

6.  (a  +  2  5  +  3  0^.  14.    *(\-x  +  x^-a?y. 

7.  (2a-35  +  (?)2.  15.    '^(a^^x^-x-\-iy, 

8.  (3m-47i-5)2.  16.    *(a3-3a2_4a4.5)X 

DIVISION 

In  certain  cases  where  both  dividend  and  divisor  are 
binomials  we  are  able  to  write  the  quotients  without 
actual  division.  Such  divisions  are  possible  only  in  cases 
where  the  powers  of  the  terms  of  the  dividend  are  like ; 
that  is,  both  terms  of  the  dividend  must  be  squares,  both 
cubes,  both  fourth  powers,  etc.     We  will  consider 

*  After  squaring,  the  terms  should  be  collected. 


DIVISION  103 

I.    The  Difference  of  Two   Squares 
Since  by  Art.  71, 

it  follows  that 

_  =  a  —  b  and =a-\-b, 

a-{-o  a  —  b 

From  which  we  state  the  general  principles : 

75.  The  difference  of  the  squares  of  two  quantities  may  be 
divided  by  either  the  sum  or  the  difference  of  the  quantities. 

If  the  divisor  is  the  sum  of  the  quantities^  the  quotient  will 
be  the  difference  of  the  quantities. 

If  the  divisor  is  the  difference  of  the  quantities,  the  quo- 
tient will  be  the  sum  of  the  quantities. 

Exercise  33 
Divide  by  inspection : 

1.  a^-x^  hj  a-x.  8.  4  a2  _  1  by  2  a  - 1. 

2.  a2  _  ^  by  a-c.  9.  4  a^  _  25  by  2  «  -  5. 

3.  a^  —  m^  by  a-{^  m.  10.  9  m^  —  16  by  3  m  +  4. 

4.  x^-y^  by  X-  y.  11.  25  w^  -  81  by  5  m  -f  9. 

5.  ^_lbya;  +  l.  12.  16  m*- 36  by  4  ^2  -  6. 

6.  c2-4byc-2.  13.  100 - 81  a:^  by  10-9^:2. 

7.  m2  -  25  by  m  +  5.  14.  x^y^  -  144  by  xy  +  12. 

15.    81  a%^  -  169  c  by  9  a6  -  13  c. 
ie.    196w*-49c6  by  14m2+7<?3. 


104  SPECIAL   CASES 

n.    The  Difference  op  Two  Cubes 
By  actual  division 

a  —  0 

From  the  form  of  the  quotient  and  its  relation  to  the 
divisor  we  state : 

76.    The  difference  of  the  cubes  of  two  quantities  may  be      i 
divided  by  the  difference  of  the  quantities. 

The  quotient  is  the  square  of  the  first  quantity.,  plus  the 
product  of  the  two  quantities^  plus  the  square  of  the  second 
quantity. 

Exercise  34 

Divide  by  inspection : 

1.  a^-7^  hj  a-x.  9.  27  -  (?3  by  3  -  <?. 

2.  a^  —  (^  by  a  —  c.  10.  a%^  —  1  by  ab  —  1. 

3.  a^  —  m^  by  a  —  m.  11.  'j^y^  —  125  by  xy  —  5. 

4.  a?  —  y^  by  x  —  y.  12.  27  a^  —  64  c^  by  3  a  —  4  c. 

5.  a^-S  hy  a-2.  13.  8  a^ -21  b^  by  2  a- Sb. 

6.  0^-21  by  x-S.  14.  125a3_8  by  5a-2. 

7.  a3  -  1  by  a-1.  15.  1000  -  c^d^  by  10  -  cd. 

8.  8 -a^  by  2-2:.  16.  729-512^:3  by  9-8:r. 

III.     The   Sum  of   Two  Cubes 
By  actual  division 

■ —  =  a^  —  ab  +  o^  • 

a  -hb 


DIVISION  105 

From  the  form  of  the  quotient  and  its  relation  to  the 
divisor  we  state  : 

77.  The  sum  of  the  cubes  of  two  quantities  may  he  divided 
hy  the  sum  of  the  quantities. 

The  quotient  is  the  square  of  the  first  quantity^  minus  the 
product  of  the  two  quantities^  plus  the  square  of  the  second 
quantity. 

Exercise  35 

Divide  by  inspection : 

1.  a^-\-h^  by  a  +  h.  8.  m^  +  125  by  m  +  5. 

2.  m^  +  n^  by  m  +  n.  9.  27  +  a^  by  3  +  a. 

3.  e^-\-m^  by  (?  +  m.  10.  27  <?3  +  64  by  3  c  +  4. 

4.  d^  +  7^hjd^-x.  11.   64  2:3  +  125  by  4  :rH- 5. 

5.  0^  +  ^  by  2: +  2.  12.  272^3^4-125  by  3  2:?/ +  5. 

6.  a;3  +  27  by  a:  +  3.  13.  125  a^  +  512  by  5  a  +  8. 

7.  2j3^_64  by2;  +  4.  14.  512^3 +  72963  by  8a  + 96. 

15.  1  +  1000^:3  by  1  +  10  a;. 

16.  1000  a%^  +  64  c  by  10  a5  +  4  e, 

IV.   The  Sum  and  Difference  of  Any  Two  Like  Powers 

By  actual  division  we  may  prove  that  the  following 
binomials  may  be  divided  by  the  corresponding  binomial 
divisors.  The  different  groups  are  arranged  according  to 
the  exponents  of  the  powers,  whether  odd  or  even  num- 
bers, and,  also,  according  to  the  signs  between  the  terms. 


106 


SPECIAL   CASES 


The  Difference  of  Even  Powers, 


etc. 


may  each  he  divided  by  either  a  -\-  b  or  a  —  b. 


The  Difference  of  Odd  Powers, 

- 

.   may  each  be  divided  by  a  —  b  only. 


a-b  ^ 
a^-¥ 
a^-h^ 
a''  -b'^ 
etc. 


The  Sum  of  Odd  Powers 

a+b 
a^  +  b^ 
a^  +  6^ 
a7  +  67 

etc. 


may  each  be  divided  by  a  -\-  b  only. 


The  Sum  of  Even  Powers. 

a^  +  J2,  a*  +  &^,  a®  +  &®,  etc.,  are  not  divisible  by  either  a  +  b  or  a  — b. 

The  following  divisions  will  assist  us  in  deriving  the 
method  for  writing  the  quotients  of  any  binomial  divi- 
sions by  inspection. 

'  =a^  +  a^b  +  ab^  +  b\ 

=  a*  +  a^b  +  a%^  +  ab»  +  b*. 

=  a*  -  a^b  +  a^b^  -  ab^  +  &*. 
a-\-b 


a* 

-M 

a 

-b 

a* 

-b^ 

a 

+  b 

a6 

-66 

a 

-b 

a6 

+  66 

hiviston  107 

From  the  form  of  the  quotients  and  their  relations  to 
their  divisors  we  state : 

78.  1.  The  number  of  terms  in  the  quotient  is  the  same 
as  the  exponent  of  the  powers  of  the  dividend. 

2.  The  exponent  of  a  in  the  first  term  of  the  quotient  is  the 
difference  between  the  given  exponents  of  a  in  the  dividend 
and  divisor^  and  this  exponent  decreases  by  1  in  each  succes- 
sive term. 

3.  The  exponent  of  h  is  1  in  the  second  term  of  the  quo- 
tient, and  this  exponent  increases  by  1  in  each  successive 
term  until  equal  to  the  difference  of  the  given  exponents  of  b 
in  the  dividend  and  divisor. 

4.  If  the  sign  of  b  in  the  divisor  is  +,  the  signs  of  the 
quotient  are  alternately  4-  and  —;  if  the  sign  of  b  is  — , 
the  quotient  signs  are  all  + . 

Exercise  36 
Divide  by  inspection : 
,     a^-b^  ^    x^  +  m^  „     //8_i 

■L.     --•  6.     •  11. 


a-b 

a^  —  a^ 

a  —  x 

a^  +  y^ 

x  +  y 

m^-a^ 

m-\-x 

(?7  +  C^7 

2. 7. 12. 


13. 


4. 9.    '-—'  14 


x-^m 

a^-c^ 

a—  c 

a%^-c^ 

ab  -{-  c 

^+1 

x  +  l 

s"  +  /I 

a-1 

1-x^^ 
1  +  x  ' 

x^y^  —  z^ 
xy  —  z 

rn)  —  Yi^ 
m  —  n 

5.    .-^.  xo.   :— .  15.    ^±|2. 

c-\-  d  x  +  y  w  +  2 


108  SPECIAL  CASES 

Exercise  37 
Oral  Drill  in  Multiplication  and  Division  by  Inspection 
Give  the  fallowing  products  : 

1.  (a +  5)2.  11.  (7w  +  3?/;3)(7m-3«/2). 

2.  (a -7)2.  12.  (a  +  5)(a  +  3). 

3.  (a  +  Sxy,  13.  (a;-9)(a;-7). 

4.  (w-5n)2.  14.  (a6-6)(a5  +  4). 

5.  (3a;-2«/)2.  15.  (m-14)(w  +  7). 

6.  (bax-^hyf,  16.  (c- ll)(c  +  15). 

7.  (7?ww-ll)2.  17.  (3a  +  l)(2a  +  l). 

8.  («  +  7)(a-7).  18.  (4  2-7)(50-2). 

9.  (3a  +  5)(3a-5).  19.  (3a:+10)(2  2;-4). 
10.  (5a;-12)(5rc  +  12).  20.  (5  a  -  7)(7  a  +  10). 

Give  the  quotients  of  the  following : 
o,     -       -  oa    w^-27  ,,     64^:3  +  216 

21. 27. 33.    ■ 

8rr  +  6 
125a;3  +  343 

5a;-i-7 

125a;3_^729 

52;  +  9 


22.    -•  28.    —•  34. 


23.    -.  29.    — 35. 


aP' 

-62 

a 

-h 

a2 

-y^ 

a 

+  h 

aP 

-16 

a 

+  4 

m^ 

-36 

m 

+  6 

977Z2-64 

3 

w  +  8 

7^ 

-8 

24. 30.    — 36. 

w  +  6  ^x  —  o 


m-3 

(73-125 

^-5 

8-27^:3 

2-3a: 

64:r3-125 

4:X-5 

8^34.1 

2a  +  l 

27  w3  + 125 

a  —  x 

m^  —  n^ 

m—  n 

<^  +  d^ 


32.         ^        ' 38. 

x-2  3m  +  5  c?  +  (^ 


CHAPTER   IX 

FACTORING.     REVIEW 

'•  79.  When  an  algebraic  expression  is  the  product  of  two 
or  more  expressions,  each  of  these  latter  quantities  is  called 
a  Factor  of  it. 

Thus,  a%  =  a  X  a  X  b. 

And  the  factors  of  a%  are  a,  a,  and  b. 

80.  Factoring  is  the  process  of  separating  an  expression 
or  quantity  into  its  factors. 

Under  separate  groups  we  will  now  consider  the  types 
of  expressions  whose  factors  are  readily  obtained. 

WHEN  EACH  TERM  OF  AN  EXPRESSION  HAS  THE  SAME  MONOMIAL 
FACTOR 

Let  US  compare  the  following  expressions : 

(1)  10  -f  15  +  20  =  5(2  H-  3  +  4).  A  factor  "  5  "  in  each 
term. 

(2)  a3+2a2  +  4a=:a(a2  +  2a  +  4).  A  factor  "a"  in 
each  term. 

In  both  expressions  we  have  divided  each  term  by  the 
factor  common  to  the  terms. 

The  factors  that  make  up  the  expression  are,  therefore, 

The  common  factor  and  the  quotient  obtained  hy  dividing 
the  expression  hy  it. 

109 


110  FACTORING.      REVIEW 

Applying  the  principle  we  have  the  following : 
a2  +  a=rt(a  +  l). 
5a;8  +  10a;=  6x(x^  +  2). 
3  ms  +  9  m2  -  12  m  =  3  m(m^  +  3  m  -  4). 
da%-6  am  +  9  a&3  =  3  ah^cP'  -  2  a6  +  3  ft^). 

Exercise  38 
Factor : 

1.  5a+10.  6.  lOir-15^  +  20. 

2.  7^-35.  7.  16m-247i  +  8. 

3.  12  a -15.  8.  11  am +  22  an -11, 

4.  14m -26.  9.  lax  —  liaz-\-21a. 

5.  4a-85  +  16.  10.  lS  +  9n-21m, 

11.  5  a6<?  +  10  abx  +  15  abi/. 

12.  5  2;  —  20  a:^  -\-SOxz  —  40  a:?/2. 

13.  a2  +  a.  19.  15m^n-20mV-S6mnK 

14.  a3_^2_^.  20.  na^x  +  51a^x^-d^a2^. 

15.  m%  +  M7i2.  21.  27  w%— 9m%2  4.l8w%3. 

16.  0^1/  —  xy-\-  xy^.  22.  13  a*  —  39  a^^  —  ^b  a^n. 

17.  a^i^  —  ^252  _|_  ^253^  23.  a^x^y  —  aaPy^  —  ax^y^. 

18.  4a«-12«2+l6a.  24.  a^mH^  -  ahnhi^  -  a^whiK 

25.  a^J*  -  2  ^352  ^  3  ^253  _  4  ^455. 

26.  rn^x  —  3  w^a:  +  2  m^a;  —  5  mx, 

27.  4a46  +  5a3j2_8^253^2a54. 

28.  a^m  —  2  a^Jrw  +  aJ^^  _  ^^3  _  2  ^77^2  _  ^^^ 


TRINOMIAL  EXPRESSION  111 

TRINOMIAL  EXPRESSIONS 

I.  The  Perfect  Trinomial  Square 

81.  A  Perfect  Square  is  a  product  of  two  equal  factors. 

82.  A  Square  Root  of  a  perfect  square  is  one  of  its  two 

equal  factors.  (Square  root  is  indicated  by  the  sign  y'. 
Thus,  V16  =  4.) 

By  Arts.  69  and  70  : 

(a  -  6)2  =  a2  -  2  a&  +  h^. 

Each  resulting  expression  is  a  perfect  trinomial  square. 
In  each  product 

The  first  term  is  a  perfect  square.  1  ^^    -, 

The  last  term  is  a  perfect  square.  J 

The  middle  term  is  twice  the  product  of  the  square  roots 

of  the  square  terms. 
The  sign  of  the  middle  term  is  +  or  —  according  as  the 

sign  of  the  second  term  of  the  given  binomial  was  + 

or  — . 

These  four  elements  are  the  conditions  necessary  for  a 
perfect  trinomial  square.     Thus  : 

a2  +  10  a  +  25.  x^  -2^x  +  144.  x'^y^  +  8  xy  +  16. 

,^^2  _  18  m  +  81.        *  4  a2  +  12  a&  +  9  h\         9  c^  -  42  c^  +  49. 

are  perfect  trinomial  squares,  for  each  trinomial  has  two 
positive  square  terms  with  a  second  term  twice  the  product 
of  the  square  roots  of  the  square  terms. 


112  FACTORING.      REVIEW 

To  factor  a  perfect  trinomial  square  : 

If  necessary^  arrange  the  terms  in  ascending  or  descending 
order. 

Take  the  square  roots  of  the  first  and  last  terms^  and  con- 
nect them  with  the  sign  of  the  middle  term. 

The  resulting  binomial  is  one  of  the  two  required  equal 
factors. 

Examples :  Factor  a'^ -\- 12  a  +  S6, 

The  square  root  oi  a^  =  a. 

The  square  root  of  36  =  6. 

The  sign  of  the  middle  term  is  +  • 

Therefore,  a  +  6  is  one  of  the  two  required  factors. 

Or,  a2  +  12  a  +  36  =  (a  +  6)2. 

Factor  9  a'^ -{- 25  2^  -  SO  ax. 

Rearrange  thus,  9  a^  —  30  ax  +  25  x^. 

The  square  roots  of  the  first  and  last  terms  are  3  a  and  5  x. 

The  sign  of  the  middle  term  is  — . 

Hence,  9  a^  _  30  ax  +  25  a:2  =  (3  a  -  5  xy. 

Exercise  39 

1.  a^  +  6a-{-9.  9.  a^^-{-Uah -^4:9. 

2.  a2+8«4-16.  10.  hV-24:bx-\-lU, 

3.  a2  +  10a+25.  M..  64. -\- c^cP  -  IQ  cd. 

4.  a2_i0a  +  25.  12.  mV -\- SQ  mn  +  S24:. 

5.  ic2-14a;+49.  13.  x^yh'^ -40  xgz +  ^00. 

6.  m2+20m  +  100.  14.  a^^e^  -  12  ahc+S6. 

7.  a2+81-18a.  15.  a^mV -[-SS  amn -{- S61, 

8.  c2-30(?  +  225.  16.  a^-6ah-]-9h^. 


TRINOMIAL  EXPRESSION  113 

17.  m^  -  12  mn-\-S6n^.  24.  16^2 -24  m +  9. 

18.  2^-^lSxt/  +  Slf.  25.  9c2-42c  +  49. 

19.  4a2_4ttw  +  m2.  26.  25 a2  +  60 a6  +  36 52. 

20.  dx^-\-QxZ  +  z\  27.    16  m2- 72  7710^+81  2^2. 

21.  SCX  +  X^-^IQC^,  28.    36  2^  +  1082:2^2  4.  81  ^4. 

22.  4:X^-12x+9,  29.  49^252 4. 140 a6c+ 100 (?2. 

23.  30  a +  25 +  9^2.  30.  81  a*-198  a2a^+121a^. 

II.    The  Trinomial  whose  Highest  Power  has  the 
Coefficient  Unity 

By  Art.  72 : 

(a  +  2)(a  +  3)  =  a2  +  5a  +  6. 

The  terms  of  the  product  are  obtained  as  follows : 

a^  is  the  product  of  a  by  a. 
+  5  a  is  the  product  of  (2  +  3)  by  a. 
+  6  is  the  product  of  2  by  3. 

From  which  we  conclude : 

The  first  and  last  terms  of  the  trinomial  are  products  of 
the  first  terms  and  the  second  terms,  respectively,  in 
two  required  binomials.  The  middle  term  of  the  tri- 
nomial is  the  product  of  the  common  literal  term  of 
the  binomials  by  the  sum  of  the  numerical  terms. 

It  remains  therefore,  in  determining  the  factors  of  a 
trinomial  in  this  form,  to  write 

a2  +  5a  +  6=:(a+    )(a  +    ), 

the  missing  terms  being  the  two  factors  of  6  whose  sum  is 
5  (i.e.  2  and  3). 

F.    H.  S.  FIRST  TEAR  ALG.  — 8 


114  FACTORING.      REVIEW 

Similarly : 
a2  +  10  a  +  16  =  (a  +  ?)(a  +  ?).     (The  factors  of  16  whose  sum  is  10.) 
x2  -  9  a;  +  20  =  (a;  -  V)  (x  -  ?) .     (The  factors  of  +  20  whose  sum  is  -  9.) 

In  the  following  exercise  the  W  possible  cases  of  ex-     ' 
pressions  formed  by  binomials  having  like  or  unlike  signs 
in  their  second  terms  are  separately  considered.  ] 

Exercise  40 

(1)  The  third  term  + .      The  second  term  + . 

The  signs  of  the  last  terms  of  both  factors  +.  *•   j 

Illustration :  x^ -h  S  x  +  15  =  (a:  +  5)  (x  +  3). 

Factor  : 

1.  ^2  +  3  ^  4.  2.  €.  2:2  +  12  rr  +  32. 

2.  a2  +  5  a  +  6.  7.  a^  +  19  a  +  48. 

3.  a^-\-l  a +  12.  8.  c2  +  20  c  +  36. 

4.  52  _^  8  5  _^  15.  9.  m^-^Slm-h  58. 

5.  w2  +  12  w  +  20.  10.  x^-{-14:x  +  48. 

(2)  The  third  term  + .      77ie  second  term  — . 
The  signs  of  the  last  terms  of  both  factors  — . 

Illustration  :  a;^  -  8  a:  +  15  =  (a:  -  5)(a:  -  3). 

11.  a2  -  4  a  +  3.  16.  ^2  _  14  ^  _^  24. 

12.  w2  -  5  m  +  4.  17.  c^  -lie  -^  24. 

13.  2^2  _  13  ^  4.  12.  18.  2:2  _  10  a;  +  24. 

14.  a^-Sa-^  15.  19.  m^-15m-h  36. 

15.  52  _  11  5  +  28.  20.  c2  -  15  c  4-  44. 


TRINOMIAL   EXPRESSIONS  115 

(3)   The  third  term  — .  The  second  term  either  +  or  —. 

The  signs  of  the  last  terms  of  the  factors  unlike^  the 

greater  last  term  having  the  same  sign  as  the  second  term  of 
the  given  trinomial. 

Illustration  :  a:^  +  2  a;  -  15  =  (a:  +  5)  (a:  -  3). 

a;2-2a;-15=  (a;  -  5)  (a:  +  3). 

21.  x^  -\-  X  -  6,  28.  m^  -11m-  60. 

22.  m^  +  m-  20.  29.  c^-15c  -  34. 

23.  a^-  a-  30.  30.  m^+11  m-  60. 

24.  x^+Zx-  10.  31.  ^2  _  23  a  -  78. 

25.  52  _  5  5  _  g,  32.  ^2  _  9  ^  _  112. 

26.  d^  -ba-  14.  33.  z2  _^  13  ^  _  140. 

27.  a2  +  8  a  -  20.  34.  y'^  -  (d  y  -  135. 

35.  a2  4-  14  a  +  24.  45.  :r2  -  11  a:  +  24. 

36.  2^2  _  9  a;  _  22.  46.  x^-  bx-  24. 

37.  c2  +  7  (?  -  60.  47.  a2  -f  17  «  +  60. 

38.  x^-l\x-  26.  48.  39  -  16  ?/  +  y2. 

39.  10  +  7  :r  +  a:2.  49.   m^  -  15  m  -  76. 
•40.  28  -  16  w  +  w2.  50.  c2  -  72  +  c. 

41.  :r2  -  2;  -  132.  51.  x^  -\-SSx-  70. 

42.  110  -  53  ^  -  x^  52.  m2  -  8  m  -  84. 

43.  y^  -15y  -  54.  53.    :?:2  _,_  2  :z;  _  120. 

44.  c^  +  21c-  46.  54.   ip'-y-  210. 


116  FACTORING.      BEVIEW 

III.  The  Trinomial  whose  Highest  Power  has  a  Coefficient 
Greater  than  Unity 
By  Art.  73  : 

(a  +  3)(2  a  +  1)  =  2  a2  +  7  a  +  3. 

The  terms  of  the  product  are  made  up  as  follows  : 

2  a^  is  the  product  of  a  by  2  a. 
+  3  is  the  product  of  +  1  by  +3. 

+  7  a  is  the  sum  of  the  cross-products  of  a  by  1  and  2  a  by  3, 
or,  a  +  (3  X  2  a)  =  a  +  6  a  =  7  a. 

Or,  applying  the  principle  to  any  pair  of  binomials,  we 
may  state  the  general  conditions  governing  their  product 
thus  : 

The  first  term  is  the  product  of  the  two  first  terms  of 
the  binomials. 

The  last  term  is  the  product  of  the  two  second  terms  of 
the  binomials. 

The  middle  term  is  the  algebraic  sum  of  the  cross-prod- 
ucts of  the  first  and  second  terms  of  the  given  binomials. 

Knowing,  therefore,  the  process  by  which  each  term  was 
obtained  in  multiplication,  we  reverse  the  reasoning  and 
find  the  numbers  from  which  the  terms  were  formed. 

Thus,  to  factor  2  a^  +  7  a  +  3, 

we  seek,  1st,  Two  numbers  whose  product  is  2  a^. 

2d,    Two  other  numbers  whose  product  is  +  3. 
3d,    That  particular  arrangement  of  these  selected  numbers 
the  sum  of  whose  cross-products  is  +  7  a. 

The  following  discussion  will  show  that  a  particular 
arrangement  of  the  selected  term-factors  is  necessary  to 
obtain  the  middle  term  of  the  given  trinomial. 


I 


TRINOMIAL   EXPRESSIONS  117 

I       Factor  6^2  +  19^  +  15. 

Since  all  the  signs  of  the  trinomial  are  +,  the  signs  of  the  terms  of 
the  binomial  factors  will  be  + . 

Factors  of  6  :         Factors  of  15 ; 

2  3 

3  5 

The  sum  of  the  cross-products,  (3  x  3)  +  (5  x  2)  =  19. 

Hence  the  factors  of  6  a^  +  19  a  +  15  =  (2  a  +  3)  and  (3  a  +  5). 

To  show  that  this  one  pair  of  factors  and  this  particular  arrangement 
of  the  numerical  coefficients  are  the  only  possible  selections  that  will 
give  by  multiplication  both  end  terms  and  the  middle  term  of  the 
given  trinomial,  let  us  examine  the  results  from  other  selections. 

Suppose  we  had  selected 

Factors  of  6 :        Factors  of  15 : 
6  15 

1  1 

The  sum  of  the  cross-products,  (1  x  15)  +  (1x6)=  21. 
Hence,  this  arrangement  is  manifestly  wrong. 

Again,  Factors  of  6 :        Factors  of  15 : 

6  1 

1  15 

The  sum  of  the  cross-products,  (1  x  1)  -f  (15  x  6)  =  1  +  90  =  91. 
And  the  arrangement  fails  to  produce  the  required  middle  term. 

Therefore,  we  conclude  : 

In  every  case  we  seek  the  particular  pairs  of  factors  of  the 
first  and  last  terms  of  the  given  trinomial^  the  algebraic  sum 
of  whose  cross-products  shall  he  the  middle  term. 

The  following  exercise  is  so  arranged  as  to  assist  the 
beginner  in  determining  the  different  possibilities  that  may 
arise  from  like  and  unlike  signs  in  the  binomial  factors 
involved. 


118  FACTORING.      REVIEW 

Exercise  41 

(1)  The  third  term  +.      The  second  term  +. 
The  signs  of  the  last  terms  of  both  factors  +. 

Factor  : 

1.  2  «2  +  3  ^  +  1.  5.  8  a2  +  22  a  +  15. 

2.  3  ^2  +  5  a  4-  2.  6.  6  ^2  +23  a  +  20. 

3.  4  a2  +  8  a  +  3.  7.  9  ^2  +  21  a  +  10. 

4.  6  a2  +  17  a  +  12.  8.  25  a2  +  35  a  +  12. 

(2)  The  third  term  +  •      The  second  term  — . 
The  signs  of  the  last  terms  of  both  factors  — . 

9.  6a^-lx-h2,  13.  12  a;2  _  23  re  +  10. 

10.  4  2^2  -  16  2;  +  15.  14.  Qx^-Ux-\-  35. 

11.  6x^-l^x  +  6.  15.  28  2^2  _  45  ^  +  18. 

12.  10  2^2  _  41  ^  _^  21.  16.  15  2^2  _  59  2;  +  56. 

(3)  The  third  term  — .     The  second  term  either  -^  or  —. 
The  signs  of  the  last  terms  of  the  factors  unlike^  the 

greater  cross-product  having  the  same  sign  as  the  second  term 
of  the  given  trinomial. 


17. 

6^2+7^_5. 

24. 

30c2-ll(?-30. 

18. 

102:2_3^_13. 

25. 

9^2-15^-50. 

19. 

15m2-7m-4. 

26. 

12  2:2_40a,_63. 

20. 

122;2_i3^_14, 

27. 

1022  4.;2_24. 

21. 

16  a2  4. 18  a -9. 

28. 

21a2+65a-44. 

22. 

20^>2_276-14. 

29. 

42a2-25a-60. 

23. 

21a2_23a-20. 

30. 

20m2-37?w-18. 

TRINOMIAL  EXPRESSIONS  119 

Exercise  42 
Factor:  Miscellaneous  Factoring 

1.  a2_  14^^49.  '      15.   2b-10x  +  3^. 

2.  4a24_7a  +  3.  16.   a;2  +  49a;  +  48. 

3.  5tt2_35^.  17.   128-24c  +  c2. 

4.  2a2_7^_^6.  18.   15a^-a:-28. 

5.  2^2  _|.  13  ^  +  30.  19.    a;2-16a:  +  48. 

6.  9:r3_36^2_|_io8a^.  20.   6'2t^  -  8  (?d;  +  16. 

7.  36-182:  +  a;2.  21.   m27i2-8m7i-48. 

8.  a;2_i4^4.48.  22.    a%'^-10a%'^-lba%K 

9.  4(?2-f.9c  +  2.  .    23.   2^ -19 2: +48. 

10.  6?2-19a  +  60.  24.   3m2-25w-18. 

11.  Uac  +  l^hc-22cd.         25.   49^4-112^2/1  +  64^2. 

12.  36^2-60^  +  25.  26.    12  -"Imn  +  mH^ 

13.  a:2_iia;_180.  27.    24  a6  +  9  ^2  +  16  62. 

14.  a^-mx-nO.  28.   a:2_26:r+48. 

29.  11  m^n^  +  22  ^2^^  -  77  m^yi^. 

30.  a2_i2a-28. 

31.  52  wV  +  39  ^271  —  26  mn^  —  13  mhi^. 

32.  -18  0^  +  2:2  +  81.  35.    ^m^-50mn  +  7n^. 

33.  a;2_i3^_48.  35.   2:2-222:-48. 

34.  4  2;2+17a:  +  4.  37.   4:n^i-9x^-l'2nx. 


120  FACTORING.      REVIEW 

BINOMIAL  EXPRESSIONS 

I.  The  Difference  of  Two  Squares 

By  Art.  71 : 

(a  +  b)(a-b)  =  a^-b^ 

Hence,  the  factors  of  a^  —  b^  are  (a  +  b)  and  (a  —  5). 
In  like  manner  : 

a^  -  x^  =  (a -\- x)(a  -  x). 

a2-9  =  (a  +  3)(a-3). 

4a2-25  =  (2a  +  5)(2a-5). 

Hence,   in    general,   to   factor   the   di£Perence   of    two 
squares : 

Extract  the  square  roots  of  the  square  terms. 

One  factor  is  the  sum  of  their  square  roots. 

The  other  factor  is  the  difference  of  their  square  roots. 

Factor  :  Exercise  43 

1.  a^-b^.  11.  4a2_9.  21.  *a^  -  xK 

2.  a^-x\  12.  9^2-25.  22.  *a:*-/. 

3.  a^-1.  13.  16^2-9.  23.  *a^-ie. 

4.  m2-l.  14.  4a;2-25.  24.  *m*  -  81. 

5.  a2_4.  15.  9^2-64.  25.  *16m4-8l2;*. 

6.  2^-9.  16.  a2_9  52.  26.  9a262_49^^2. 

7.  ?/2_i6.  17.  4a2-49a;2  27.  16a2^,2_9^^2,,2, 

8.  ^2-25.  18.  16  2^2  _  25  ^2,  28.  121  a^b^-c\ 

9.  2^-49.  19.  36  0:2-81^2.  29.  25a;4/-862io. 
10.  w2-81.  20.  49^2-100^2.  30.  144^10-169. 

*  Three  factors. 


BINOMIAL  EXPBESSIONS  121 

31.    (a  +  hy-c^=  (a-\-h-hc)(a  +  b-c).     Result. 

32.  (m-ny-x^     34.   (a-2f-x^.      36.   16(«  +  2)2-9. 

33.  ((?4-t?)2-l.       35.  iQa  +  hy-c^    37.  36((?  +  2)2-49. 

38.    a2_(5_|.^)2. 

=  (^a-\-b-\-c)(a-b-c).    Result. 

39.  m^-(n  +  iy.  43.  25-9(a+J)2. 

40.  a2_(^_i)2.  44,  36a2_25(a  +  l)2. 

41.  16-(a  +  ^)2.  45.  49w2-9(w-l)2. 

42.  25-(3  +  2;)2.  46.  64ic2«/2_25(ajy+l)2. 

II.   The  Difference  of  Two  Cubes 

83.  A  Perfect  Cube  is  a  product  of  three  equal  factors. 

84.  A  Cube  Root  of  a  perfect  cube  is  one  of  its  three 
equal  factors. 

By  Art.  76:  2i,i^  =  „.  +  <,j  +  j,. 

a  —  b 

From  which  the  factors  of  a^  —  b^  are  (a  —  b)  and  (a^  +  ab  +  b^). 

Similarly  ;        a^  -  x^  =  (a  -  x)  (a^  +  ax  +  x^). 
c3_8  =  (c-2)(c2+2c  +  4). 
27^3  -  64  =.  (3  m  -  4)  (9  m^  +  12  m  +  16). 

Hence,  in  general,  to  factor  the  difference  of  two  cubes  : 

One  factor  is  the  difference  of  the  cube  roots  of  the 
quantities. 

The  other  factor  is  the  sum  of  the  squares  of  the  cube 
roots  of  the  quantities  plus  their  product. 


u 

52 

FAC 

'TOEING,      REVIEW 

Factor : 

Exercise  44 

1. 

a^  -  ¥. 

8. 

2^3-125. 

15. 

8^3-27. 

2. 

a^  —  a^. 

9. 

27 -m3. 

16. 

27  -  64  m3. 

3. 

m^  —  n^. 

10. 

64-2:3. 

17. 

125^3-64^3. 

4. 

e^-1. 

11. 

a%^-U. 

18. 

216  2^3-27. 

5. 

^3-8. 

12. 

mV  -  125. 

19. 

27^363-512. 

6. 

0^-21. 

13. 

27  -  a^y\ 

20. 

8  m3- 343^13. 

7. 

m3-64. 

14. 

64  —  mV. 

21. 

125  2:3_  729^^3. 

III.   The  Sum  of  Two  Cubes 


By  Art.  77:        ?^=a^-ah  + 


From  which  the   factors    of   a^  +  ^3   are    (a  +  h)    and 
ia^-ab  +  h^). 

Similarly  :         a^  +  x^=(a  +  x) (a^  -  ax  +  x^). 
c8  +  8=(c  +  2)(c2-2c  +  4). 
125  +  a:3  =(5  +  a;)(25  -  lOx  +  x^). 
27  m3  +  64  =  (3  m  +  4)(9  m^  -  12  m  +  16). 

Hence,  in  general,  to  factor  the  sum  of  two  cubes : 

One  factor  is  the  sum  of  the  cube  roots  of  the  quantities. 
The  other  factor  is  the  sum  of  the  squares  of  the  cube  roots 
of  the  quantities  minus  their  product. 

Factor:  Exercise  45 

1.  2^3  +  ^3.  4.    ^3+8.  7.    (^  +  125. 

2.  a^+m^  5.   (?3_^27.  8.   27  +  2^8. 

3.  2:3^1.  ^.   a3_|_64,  9.   64-1-^3. 


EXPRESSIONS  HAVING  FOUR   TERMS  123 

10.  wV+64.         13.   M-^M^  16.   8^3+343^. 

11.  A3 +  125.        14.   8^3+27  63.  17.   512^3+27^^. 

12.  27H-a353.  15.   216^3  4- 27.         18.   1000  2^3^729. 

Exercise  46 
Factor  •  Miscellaneous  Factoring 

1.  a^-9x^.  11.   64^3+125.  21.  Q4:c^-}'1, 

2.  a3-8m3.  12.   25-64m2.  22.  81  c*^^*- 36  ic^. 

3.  a3+27.  13.   216-27(?3.  23.  729a^-Sc^ 

4.  c3  -  27  a;3.  14.   4  m*  -  9  a;*.  24.  64  a;3  -  125  ^\ 

5.  2:3  4_  125.  15.   8w3n3-125.  25.  64a:6_i. 

6.  4  77i2_9.  16.   25  2:6 _  9.  26.  64  2:8_i. 

7.  1-64  2:3.  17.    272:6-8/.  27.  512  2:6  _  27. 

8.  27  2:3  +  8.  18.   49m6-9  7i6.  28.  1292^-125^. 

9.  125^3-8.         19.   8^6 -125  53.  29.  1000  2:3  _  27  2/3. 
10.   25  a2- 36  2:2.      20.   64  m9- 27  7^9.  30.  1728  2:3_  1331. 

EXPRESSIONS  HAVING  FOUR  TERMS 

Certain  common  expressions  having  four  terms  can  be 
so  grouped  as  to  come  under  types  that  have  already 
been  considered. 

I.   The  Grouping  of  Terms  to  show  a  Common 
Binomial  Factor 

In  the  expression  ax  +  bx  +  ex,  the  factor  "  x  "  is  common  to  the 
terms.    Factoring, 

ax  +  bx  -{•  ex  =(a  +  b  +  c)x. 


124  FACTORING.      BEVIEW 

Similarly,  in  a(x  -\-  y) -\-  h{x  +  ?/)+  c{x  +  y),  the  binomial  factor 
"  (a;  +  y)  "  is  common  to  the  terms.     Factoring, 

a{x  +  y)+  h{x  +  y)-\-c{x  +  ij)  =  {a  +  &  +  c)(a;+ ?/). 

In  the  above  illustration  the  common  binomial  factor  is 
readily  seen,  but  in  many  expressions  it  is  necessary  to 
make  a  careful  examination  of  the  given  terms  before  a     , 
particular  binomial  can  be  found.     In  such  cases  we  find,     I 
by  inspection,  what  terms  hear  to  each  other  the  same  rela- 
tion; and  then  we  obtain  a  common  binomial  factor  by  the     _, 
necessary  rearrangement  and  grouping.     Thus : 

Factor  ax -{- ay  -\- hx  ■\-  hy. 

It  will  be  noted  that  "  a  "  is  common  to  the  first  two  terms,  and 
that  "&"  is  common  to  the  last  two  terms.  Furthermore,  if  the 
common  factor,  a,  is  taken  out  of  the  first  two  terms,  the  same  expres- 
sion results  as  when  the  common  factor,  5,  is  taken  from  the  last  two 
terms. 

Hence,  the  process : 

ax  +  ay  -\- hx  ■{  hy  =  (ax  +  ay)  +  (bx  +  hy) 
=  a(x-{-y)  +  b(x-^y) 
Collecting  coeflacients,  =(a  +  h)(x  +  y).     Result. 

Sometimes  we  make  use  of  the  parenthesis  preceded  by  the  minus 
sign  in  order  to  group  the  terms  so  as  to  show  a  common  binomial 
factor.     Thus : 

Factor  ax  +  ay  —  bx  —  hy. 

ax  -\-  ay  —  bx  —  by  =  (ax  +  ay)  —  (bx  +  by) 

=  a(x  +  y)-b(x  +  y) 

Collecting  coefficients,  =(a  —  b)(x  +  y).     Result. 

Many  examples  of  this  type  admit  of  more  than  one  way  of  group- 
ing terms.  But  the  final  result  is  unchanged,  no  matter  what  may 
have  been  the  grouping  for  a  common  binomial  factor.  The  change 
is  in  the  order  of  the  resulting  factors. 


EXPliESSIONS  HAVING  FOUR    TERMS  125 

Factor:  Exercise  47 

1.  ac -\- be -\- ad -\-  bd.  11.  m^—  2m-\-  mx  —  2  x. 

2.  am  +  mx -\-an  +  nx.  12.  2c^  —  6c-{-cd  —  Sd. 

3.  ac  —  bc+  ad—  bd.  13.  a^  -\-  a^  +  a  -^  1. 

4.  ayn  —  mx  -{-an  —  nx.  14.  a?^  +  a;^  +  a;  +  1. 

5.  a^  +  ab -\- ac -\- be.  15.  ^3_^  3  ^2_|_  2  ^  + 6. 

6.  e^  -\-  ex  -\-  e?/  -i-  xy.  16.  2:^  +  a:^  —  re  —  1. 

7.  a2  4.  ^5  _^  <^  4.  5.  •  17.  ^4  _|_  ^  _  ^  _  1, 

8.  ^2  4-  aa;  +  «  +  a:.  18.    m^  4-  2  m^  —  9  m  —  18. 
^.    a^-ax^a-x.  \^.    e'^-\-2(f^-\-e-\-  2. 

10.   a^-ab^-2a-2b.  20.    a*  +  3  a^  -  27  a  -  81. 

II.  The  Grouping  of  Terms  to  form  the  Difference  of 
Two  Squares 

By  multiplication : 

(a  +  &  +  c)  (a  +  &  -  c)  =  «2  +  2  a&  +  Z>2  _  c2. 
(x  +  a  +  2)  (x  +  a  -  2)  =  x2  +  2  aa:  +  a2  _  4. 
{x  -  y  ^  z){x  -  y  -  z)=  x^  -  1  xy  -^  y'^  -  z^. 

Each  result  has  four  terms.  Three  of  the  terms  are  per- 
fect squares  and  the  fourth  term  is  twice  the  product  of  the 
square  roots  of  two  of  the  square  terms.  With  these  facts 
in  mind,  the  student  will  carefully  factor  the  following 
expressions  of  similar  form. 

Factor  a^^-2  ab+b'^- x^. 

a2  +  2  a6  +  62  -  a;2  =  (a2  +  2  a&  +  &2)  _  x"^ 
=  (a  +  by-x^ 
=  (a-\-b  +  x)(a  +  h  —  x).     Result. 


126  FACTOBING,      REVIEW 

Factor  a:^ -h'^-2hc- (^. 

=  [a  +  (6  +  c)][a-(6  +  c)] 

=  (a +  &  +  c)(a-6-c).     Result. 

Factor  a;^  _^  ^2  _  ^2  _  2  ^a?. 

a;2  +  a2  _  ^2  _  2  ax  =  (x2  -  2  ax  +  a2)  -  ^2 
=  (a;-a)2-z2 

=  (x  —  a  +  2)  (x  —  a  —  z).     Result. 

It  will  be  clear  to  the  student  that  the  grouping  is' 
simply  a  matter  of  determining  what  three  terms  make  up 
the  desired  trinomial  square.  The  term  that  is  not  a  square 
is  the  hey  to  the  grouping  arrangement,  for  its  literal  factors 
show  the  square  terms  with  which  it  must  be  grouped. 

It  may  be  of  help  to  the  student  to  familiarize  himself 
with  still  another  method  by  which  the  proper  grouping 
may  be  obtained.  From  the  signs  of  the  square  terms  we 
derive  the  following  : 

(1)  When  only  one  given  square  term  is  plus  it  is 
written  firsts  and  the  other  three  terms  are  inclosed  in  a 
pareiithesis  preceded  hy  a  minus  sign. 

(2)  When  only  one  given  square  term  is  minus  it  is 
written  last.,  and  the  other  three  terms  are  written  first  in 
a  plus  parenthesis. 

Factor :  ExerclBe  48 

1.  a^^^ab-{-h^-m\  4.  4:<^- ^cd-\- d? -1. 

2.  m^  +  ^mn+n^-x^.  5.  4:a^-12ax-\-9 x^-25. 

3.  x^-2xy-^y^-z^  6.  a^ +  b^- (^-{.2aL 


GENERAL  AIDS  TO  FACTORING  127 

8.  m2  — 7i2+2m  +  l.  15.   4  2:2-^2  — 2a  — 1. 

9.  a^  +  2/2  —  ^2  _  2  a^y.  16.   m^ -\- 10  xi/ —  x^  —  25 1/\ 

10.  a:4  +  a:2-4-2a;3.  17.  e^  _  ^6  -  d^-12d, 

11.  a2-62_25c-c2.  18.  4a2  +  l-4a:2_4^. 

12.  w2-ri2-27i-l.  19.  -2rz;  +  l  +  2;2_a:4. 

13.  a;2-/  +  2y-l.  20.  6<?  +  9a2-l-9tf2. 

REPEATED  FACTORING 

The  process  of  factoring  must  frequently  be  applied 
more  than  once  before  an  expression  is  resolved  into  its 
prime  factors.  In  the  following  review  it  is  first  neces- 
sary that  each  example  be  carefully  examined  to  determine 
under  which  type  it  belongs.  Then,  having  determined 
that  a  particular  type  will  apply,  it  remains  to  carefully 
examine  resulting  factors  to  see  whether  or  not  they  are 
themselves  to  be  refactored.  From  the  different  type- 
forms  that  have  been  presented  we  are  able  to  make  the 
following  classification  of  expressions  that  we  have  learned 
to  factor.  Basing  our  conclusions  directly  upon  the  num- 
ber of  terms  in  a  given  expression,  we  are  able  to  make 
the  following  helpful  hints  for  the  factoring  of  miscel- 
laneous and  unclassified  forms. 

GENERAL  AIDS  TO  FACTORING 

First.  Determine  if  each  term  has  a  common  monomial 
factor.  If  so,  remove  it,  and  apply  the  further  ex- 
aminations suggested  below. 


128  FACTORING.      REVIEW 

Second.    What  is  the  number  of  terms  in  the  expression  to  he 
factored  I* 

I.    If  two  terms,  place  it  under  the  proper  binomial 
form. 

(a)  If  the  exponents  of  the  terms  are  even, 
the  coefficients  perfect  squares,  and  the 
second  term  negative  in  sign,  the  expres- 
sion is  the  difference  of  two  squares. 
(6)  If  the  exponents  of  the  terms  are  divisible 
by  three  and  the  coefficients  of  the  terms 
are  perfect  cubes,  the  expression  is  the 
sum  or  difference  of  two  cubes,  according 
to  the  sign  of  the  second  term. 
II.    If  three  terms,  place  it  under  the  proper  tri- 
nomial form.     It  is  well  to  apply  at  once  the 
test  for  a  perfect  trinomial  square,  which  test 
failing,  we  know  that  it  belongs  under  one  of  the 
two  remaining  trinomial  forms. 
III.    If  four  terms,  look  for  the  proper  grouping  that 
jvill   place   it   under  one   of  the  forms  already 
given. 

(a)  If  the  expression  has  three  perfect  square 
terms,  apply  the  grouping  test  for  the  dif- 
ference of  two  squares,  one  a  trinomial 
expression,  the  other  a  monomial. 
(5)   If  no  terms,  only  two  terms,  or  all  four 
terms  are  squares,  look  for  a  grouping  in 
pairs  that  shall  give  a  common  binomial 
factor. 
Third.    Continue  the  processes  until  the  resulting  factors 
cannot  be  refactored. 


GENERAL  AIDS   TO  FACTORING  l20 

Typical  Cases  of  Repeated  Factoring 

Factor  a3  +  10  ^2  _^  25  a. 

a8  +  10a2  +  25a  =  a{cfi  +  10a  +  25) 
=  a{a  +  5)2.    Result. 
Factor  a^  —  x^. 
a^  -  x6  =  (a3  +  x^)  (a3  _  x^) 

=  (a  +  a:)(a2  -  ax  +  x'^){a  -  x)(cfi  +  aa;  +  x^).     Result. 
Factor  a^- 13  2^2  ^36. 

a;4  _  i3a;2  +  36  =  (a:2  _  4)(a;2  -  9) 

=  (x  +  2)(a:  -  2)  (a:  +  3)(a:  -  3).     Result. 

Factor  a:^  —  a;*  +  a;^  —  ^. 
a;^  —  x*  +  x^  —  a:  =  a;[a;^  —  x^  +  x^  —  1] 

=  a;[(a;6-a;3)  +  (a;2-l)] 

=  a:[x8(a:2-l)+(a;2-l)] 

=  a:(a;8+l)(a:2-l) 

=  x{x  +  l)(a:2  -  a:  +  l)(x  +  l)(a:  -  1).     Result. 

Exercise  49 

Review.    Miscellaneous  Factoring 
^  Factor : 

1.  a2_i0a  +  25.  8.  a^-:^-Qx.  15.  m2-3w-70. 

2.  a2_  10^.^24.  9.  ^3-64.  16.  4a2_i2a  +  9. 

3.  a^  —  a.  10.  m2  —  64.  17.  a;*  —  8  a;. 

4.  2:2_42;4.4.  11.  x^j^\2x+20,  18.  3a2_i2^4.12. 

5.  2;2  +  5a;  +  4.  12.  a;2  +  9a;+20.  19.  272:5_^2. 

6.  m^-4m.  13.  a;24-2l2:4-20.  20.  4a:2_|_4^_,_l^ 

7.  a;2_^5^_g4^  14^  a;3_92;.  21.  m^+21, 

F.  H.  8.  FIKST  YEAR  ALG.  — 9 


130  FACTORING.      REVIEW 

22.  5^2-24^-5.  49.  c3-c2_c+l. 

23.  Sa^-lOx+S.  50.  a^-da. 

24.  Sm^-Sm,  51.  c3_7^_60<7. 

25.  81-18a^+a^.  52.  ^(a-^by-cf^. 

26.  642^-27.  53.  c2_^_42. 

27.  5a5-5a.  54.  Qa^-lx-S. 

28.  a2-lla-26.  55.  27 a;* -a:. 

29.  a3+10a2-24a.  56.  a* -16  J*. 

30.  a* -81.  57.  9(a  +  l)2-4. 

31.  a*  +a.  58.  c'^-  5c  +  4. 

32.  ^3  +  4  a2  +  4  a.  59.  a^  —  S  a  -\-  ax  —  S  x. 

33.  m2-<i2  4.2a-l.  60.  2x^-^x-55. 

34.  c2_^4^_21.  61.  l-a-a2  +  a3. 

35.  x^-1.  62.  125^3-  8. 

36.  b^-b^  +  b  —  1,  63.  a7_^. 

37.  a^  +  7  a^  +  10  a.  64.  m^  —  n^  ^  ^  _  ^, 

38.  a^-3a;3_p7^2.  gg,  80-5w*. 

39.  ^2-2^+1-0^2.  66.  a^^Sa^^Sa  +  1. 

40.  Qx^-X-12.  67.  (?5_81(?. 

41.  ab-{-be  —  ad  —  cd.  68.  a;^  —  13  a;2  _|_  4  ^. 

42.  (^-IQc.  69.  4a;4_92;2_62;-l. 

43.  a;3_2a;  +  a;2_2.  70.  36  m%*  -  13  ^2/12  4. 1. 

44.  m^ -\- m^ -\- m  +  1.  71.  343  a:^  —  1. 

45.  2^Jr2a^-x-2,  72.  64a:3+729?/3. 

46.  m5-18m3+81m.  73.  1- c^ -  c^-\- (^, 

47.  a2_3^_4.  74.  20 a:  +  100  +  a:2. 

48.  8a;2-24a;«/  +  18t/2.  75.  6m2-8m-8. 


GENERAL  REVIEW  131 

GENERAL  REVIEW  — GROUP  III 

THE  SIMPLE  EQUATION  -  AXIOMS 

The  same  quantity  may  he  added  to  or  subtracted  from  both 
members  of  an  equation. 

Both  members  of  an  equation  may  be  multiplied  or  divided 
by  the  same  quantity. 

TRANSPOSITIONS 

A  term  may  be  transposed  from  one  member  of  an  equation 
to  the  other  member  provided  its  sign  be  changed. 


PROBLEMS 

To  solve : 

(a)  Represent  the  desired  unknown  value  by  x. 

(b)  State  the  given  conditions  of  the  problem  in  terms  of  x. 

(c)  Write  the  equation  that  fulfills  the  conditions  and  solve 
for  X. 

SUBSTITUTION 

In  finding  Numerical  or  Literal  Values 

Replace  the  given  literal  factors  in  the  terms  by  the  given 
values.     Perform  all  operations  and  simplify  the  result. 

In  USING  Formulas 

Substitute  the  known  values  and  the  necessary  constant 
values.     Solve  the  formula  for  the  required  element. 

In  VERIFYING  Equations 

Substitute  in  the  original  equation  the  value  obtained  for 
the  root.  If  simplification  makes  each  member  the  same^  the 
root  found  is  correct. 


132  FACTORING.      REVIEW 

SPECIAL  CASES  IN  MULTIPLICATION  -  TYPE-FORMS 

The  Product  of  Binomials  having  Like  Terms 

(a  +  &)2  =  a2  +  2  a&  +  h\  («  _  hy  =  a^  -  2  ah -\-  b^. 

(a  +  b)(a-b)  =  a^-b^ 

Trinomials  grouped  for  Binomial  Forms 

(a  +  ft  +  c)(a  +  6  -  c)  =  [(a  +  6)  +  c]  [(a  +  6)  -  c] 
(a  +  b  +  c)(a-b  +  c)=  [(a  +  c)  +  6] [(a  +  c)  -  ft] 
(a  +  ft  -  c)(a  -  &  +  c)  =  [a  +  (&  -  c)]  [a  -  (6  -  c)] 

Binomials  having  Unlike  Terms 

(a  +  b)(a  +  c)  =  a'^  +  (b  +  c)  a  -\-  be. 

(ax  +  b)  (ex  ■}■  d)  =  acx^  +  (ad  +  &c)  x  -f  bd. 

The  Square  of  Any  Polynomial 

(a  +  &  +  c)2  =  a2  +  &2  +  c2  +  2  a&  +  2  ac  +  2  &c. 

SPECIAL  CASES  IN  DIVISION.  -  TYPE-FORMS 

The  Difference  of  Two  Squares 

=  a  -\-  0.  =  a  —  0. 

a  —  b  a  +  b 

The  Difference  of  Two  Cubes 

a  —  b 
The  Sum  of  Two  Cubes 

«i±A'=a2-aft  +  &^ 
a  +  b 


GENERAL  REVIEW  133 

The  difference  of  equal  even  powers  of  a  and  h  is  divisible 
by  both  a-\-b  and  a  —  b. 

The  difference  of  equal  odd  powers  of  a  and  b  is  divisible 
by  a  —  b  only. 

The  sum  of  equal  odd  powers  of  a  and  b  is  divisible  by 
a  +  b  only. 

FACTORING  -  TYPE-FORMS 

A  common  monomial  factor  may  occur  with  any  of  the  later 
forms. 

The  Binomial  Forms 

The  difference  of  two  squares. 
Illustration :  a^  -  b\ 

The  difference  of  two  cubes. 
Illustration  :  a^  —  h^. 

The  sum  of  two  cubes. 
Illustration :  a^  +  h\ 

The  Trinomial  Forms 
The  perfect  trinomial  square. 
Illustration  :        a^  +  2  aft  +  ft^  or  a^  -2ah  +  bK 

The  trinomial  having  the  coefficient  of  x  unity. 
Illustration  :    x^  +  bx  -\-  c.     (Where  b  and  c  may  be  any  numbers.) 

The  trinomial  having  the  coefficient  of  x  greater  than  unity. 
Illustration :  ax^  +  bx-\-  c.    (Where  a,  &,  and  c  may  be  any  numbers.) 


134  FACTORING.      REVIEW 

Expressions  having  Four  Terms 

The  polynomial  having  a  common  binomial  factor. 
Illustration :  ax  ■{■  hx  -{■  ay  ■\-  by. 

The  polynomial  containing  both  trinomial  and  monomial 
squares. 

Illustration  :  a"^ -h  2  ab  +  b^  -  c\ 

General  Suggestions  for  Factoring 

First.  Examine  the  given  expression  for  a  common  mono- 
mial factor. 

Second.  The  number  of  terms  in  the  expression  and  the 
powers  and  coefficients  will  serve  to  indicate  the  form  under 
which  it  comes. 

Third.  Apply  the  proper  separation  and  note  carefully 
whether  the  factors  found  can  be  ref adored. 

Exercise  50 

Review 
Solve  and  verify : 

1.    2x(x-\-V)-(x-iy={x-\-^\ 


2.    x-l2x-(^x-l-x^)']  =  (x-2y-l, 

5.  6(a;-l)2-3(2:+2)2-(3rr-l)(a;  +  4)  =  0. 

State  and  solve : 

6.  What  two  numbers  are  those  whose  sum  is  90,  and 
whose  difference  is  26  ? 


GENERAL  REVIEW  135 

7.  Divide  84  into  two  parts,  one  of  which  shall  be  12 
more  than  double  the  other. 

8.  Find  the  three  consecutive  odd  numbers  whose  sum 
is  129. 

9.  The  difference  between  the  squares  of  two  consecu- 
tive numbers  is  23.     Find  the  numbers. 

10.  A's  age  is  three  times  B'  s  age,  but  in  twelve  years 
A  will  be  only  twice  as  old  as  B.  Find  the  present  age 
of  each. 

Make  the  following  substitutions  : 

11.  If  a:  =  3  and  y  =  —  1,  what  is  the  value  of  (x  —  y)^ 

12.  What  is  the  value  of  (x-^a-\- 1)^  —  3  (a;  —  a)^  when 
a:  =  —  3  and  «  =  —  7  ? 

13.  Find  the  value  of  (m  +  a;)  (w  +  5  a;) — (3  m  +  2  a:  +  mxy 
when  x=  —  m. 

14.  Find  the  value  of  S  in  the  formula  S=-(a  -f  V) 
when  a  =  J,  ^  =  i\  and  n  =  21. 

15.  What  is  the  value  of  the  expression  (^x—a')(^x-\-a—y') 
—  (x—y)(x-\-a)a—  (x  —  af^  when  x  =  —  y  and  a  =  0  ? 

Obtain  the  following  required  values  by  substitution  in 
the  proper  formulas : 

16.  What  is  the  area  of  a  circle  whose  radius  is  10 
inches  ? 

17.  In  going  one  mile  how  many  turns  are  made  by  a 
wheel  4  feet  in  diameter  ? 


136  FACTOBING.      BEVIEW 

18.  How  many  feet  will  a  body  fall  in  7  seconds  ? 

19.  What  is  the  sum  of  the  first  ten  even  numbers  ? 

20.  Show  that  there  are  75.3984+  inches  in  the  circum- 
ference of  a  circle  whose  radius  is  1  foot. 

Multiply  by  inspection : 

21.  (a +  3)  (a -4).  26.  (a;  +  4)(:r-9). 

22.  (2a-l)(3a  +  2).  27.  (a^-x^J^x-iy, 

23.  (5a;-3^)(2a;-«/).  28.  (2;- 14)  (2  a; -f  5). 

24.  (a+2  6-2(?)2.  29.  (3  a -4)  (11  a +  15). 

25.  (a;-l)(a;-5).  30.  (2x  + b  y^(p  x  -  12  y^. 

Divide  by  inspection,  supplying  omissions  by  possible 
divisors : 

31.    tziA.  36.    64  +  729^3 


32.     ^^ 37. 

33.  64^. 


a-2 

27 +  a^ 

? 

64-a:3 

^-x 

27  m3  -  125 

? 

343  -  a3 

34. 39. 


35. 40. 


? 

8l2;2- 

25 

? 

27a:3_ 

343 

? 

729  a;3. 

-512 

? 

1000- 

27:1^ 

GENERAL  BEVIEW                     •           137 

Factor : 

41.  m^-16m.  49.  -12^2+^3 +  36  a. 

42.  S2^-15a^  +  9x.  50.  c^-Slc. 

43.  a^-lSa^  +  Sla.  51.  :?^  -  4  ^2  -  8  rr  +  32. 

44.  m3+7m2-18w.  52.  3^2_9^^2_22:^. 

45.  8-2a2-4a  +  a3.  53.  x^-21x. 

46.  3  a;5  _|_  17  ^  +  10  a;.  54.  ^^j*  +  a:2  _  2  a;3  _  ^2^. 

47.  36(:?;-l)2-25.  55.  9  (a +  1)2 -4. 

48.  30  a;3  +  65  a;2  _  140  2:.  56.  2;^  _  4  ^_,_^_4. 


57.  What  is  the  quotient  when  the  sum  of  x^  —  x  +  7 
and  3a:2_|_4a;_6is  divided  by  2;  +  1  ? 

58.  Solve  and  verify  (a;+ 1)(2  2:-3)  =  (2;-3)(2a:-l). 

59.  Prove   that    (a -^  x')  (a  —  x  -{-  1)  =  (a  —  :r)  (  a  +  ic) 
-\-  a  +  X. 

60.  Solve  and  verify  (2;  -  5)  (3  2:  +  7)  =  (2  a:  -  1)  (a;  -  3) 

+  x(x-^l). 

61.  Find   the    value    of    S  x  —  x {x  —  ly  —  5  x(x  -{-  4) 
—  (x  —  1)2  when  a;  =  3. 


62.  What  must  be  added  to  3a;—  [— 2a^+(^  —  5  —  a^)] 
to  make  x—  (^a  —  h  —  x^? 

63.  If  a  =  2,  1=  10,  and  r  =  —  2,  find  the  value  of  S  in 
:       the  expression :  rl  —  a 

r—  I 

64.  Show  that  —  I  is  a  root  of  the  equation,  3  (a;  —  1) 
(a;  +  l)  =  a;(3a;+4). 


138  •  .       FACTORING.      REVIEW 

65.  Find  the  value  of  (a  —  5)  (a  +  ^  —  <?)  —  («—  ^  +  0^ 
—  ahc  when  a  =  1,  ^  =  —  1,  c  =  0. 

66.  The  score  of  a  football  game  was  —  (3— [12— (3 
-2-4)]to-4-(-3  +  3-4).  What  was  the  actual 
result  of  the  game  ? 

67.  Through  what  distance  will  a  body  fall  in  3  seconds 
if  a  in  the  formula  S  =  ^  at'^  is  980  centimeters  ? 

68.  What  are  the  exact  binomial  divisors  of  the  binomial 
2:6-1? 

69.  Write  the  square  of  (a  —  3  5  +  5  c). 

70.  Find  the  value  of  (3  re  +  2)  (2  2:  -  3)  +  (2  :r  -  5)2 
when  x=  ^. 

71.  What  are  the  factors  oi  m^  —  n^  —  jj^—  2np? 

72.  If  X  equals  3,  which  is  the  greater  expression, 


2x-\Sx-(-5x-l)\  or  5x- [2  x- (^-Sx -{-!)']? 

73.  Multiply  by  inspection  (a  —  3  bx)  (7  a  +  4  hx^. 

74.  Find  the  value  oi  (Sx-4:  a}(2x+7  a^-^Sx-bay 
—  aV  when  x=2a. 

75.  If  a  =  3,  71  =  7,  and  c?  =  —  2,  what  is  the  value  of  I 
in  the  formula  l  =  a+(n—l)d? 

76.  What  are  the  factors  oi  2  xy  —  14:  y  —  ax  +  7  a? 

77.  The  divisor  in  a  certain  example  was  2x^  —  x—  3, 
the  quotient  a^-\-x—  5.     What  was  the  dividend  ? 

78.  A  man's  weight  increases  by  80  pounds,  and  he 
then  weighs  110  pounds  less  than  twice  his  original 
weight.     What  was  his  original  weight? 


GENEBAL  BEVIEW  139 

79.    What  are  the  factors  oi  b  a^ —  2i3^— 5a^? 


80.  Simplify  aH-[— a—  {  —  a  -\-  a  —  {a—  1)  \  —  a  -^  l"]. 

81.  From  what  expression  must  a^—  S  x-^1  be  sub- 
tracted to  give  a  remainder  of  x^  ? 

82.  What  is  the  sum  of  the  factors  oix^— 1  —  2  3:^ +2x7 

83.  What  is  the  result  when  (5  a:  -}- 1)  (2  a;  —  3)  is  taken 
from  (ic  +  2)  (:?:  +  1)  +  (:r  -  2)2  ? 

84.  To  what  expression  must  S  m^  —  2'm^ -[-Ihe  added 
to  produce  0  ? 

85.  What  are  the  factors  oi  x^  -  21  x^  -  4.  a^ -\- lOS  ? 

86.  A  man  receives  his  month's  salary  in  five-dollar 
and  ten-dollar  bills.  He  receives  one  more  bill  of  the 
smaller  denomination  than  of  the  larger.  If  he  was  paid 
•f  110,  how  many  bills  of  each  denomination  did  he  receive  ? 

87.  How  much  greater  than  a;  + 1  is  the  quotient  of 

88.  What  are  the  factors  oi2a^-2f-4:i/-2? 

89.  John  is  8  years  older  than  William,  and  John's  age 
is  as  much  above  6  as  William's  age  is  below  38.  Find 
the  age  of  each. 

90.  Two  men  were  paid  the  same  sum  for  a  week's 
work.  If  one  of  them  had  received  18  more  and  the  other 
$S  less,  the  one  would  have  received  just  twice  as  much 
as  the  other.     What  amount  did  each  receive  ? 


CHAPTER  X 
HIGHEST   COMMON   FACTOR 

85.  A  common  factor  of  two  or  more  algebraic  expres- 
sions is  an  expression  that  can  be  divided  into  each  without 
a  remainder. 

Thus,  a%^,  a%\  a'^b^,  and  a%^  can  each  be  divided  by  ab. 
Hence,  ab  is  a  common  factor  of  a%'\  a^b*,  a%^,  and  a^6^. 

86.  The  highest  common  factor  of  two  or  more  algebraic 
expressions  is  the  expression  of  highest  degree  that  can  be 
divided  into  each  of  them  without  a  remainder. 


Thus: 


Given, 


g^  aV)^  is   the  expression   of   highest  degree  that 

will  divide  each  of  the  four  expressions  without 
-,,     a  remainder. 


From  this  we  make  the  following  conclusion : 

87.  The  Highest  Common  Factor  of  two  or  more  expres- 
sions is  the  product  of  the  lowest  powers  of  the  factors  common 
to  the  given  expressions. 

The  abbreviation  "H.  C.  F."  is  commonly  used  in 
practice. 

THE  H.C.F.  OF  MONOMIALS 

In  the  simple  forms  of  monomial  expressions,  the 
H.  C.  F.  is  readily  found  by  inspection. 

140 


THE  n.  C.  F.    OF  POLYNOMIALS  141 

Oral  Exercise 
Find  the  H.  C.  F.  of : 

1.  a%^  and  a%^.  7.  16  m%  and  24:  nh;. 

2.  mVx  and  mnV,  8.  15a^i/^  and  20^:223^ 

3.  2m^x^  and  4m^2^.  9.  12  mV  and  ISm^/i^. 

4.  3^3(^5  and  6c(^*.  10.  IS  c^d'^m  and  2T<?3J%. 

5.  3  a'^mx^  and  9  «wV.  11.  35  a%^c^  and  42  ^352^3. 

6.  10  x^y^z^  and  15  x7/h^.  12.  51  a^^y^^  and  17  a^i/. 

13.  12  ^25^,  16a63c?,  and  20  abc^ 

14.  18  mwa;,  24  mpx,  30  /wwp,  and  36  npx. 

15.  5m^nx,  10  mn'^x,  15mnx^,  and  20w2?i2a;. 

16.  33  abx,  44  «<?a7,  55  aJ^?,  and  66  obex. 

THE  H.  C.  F.  OF  POLYNOMIALS 

When  the  given  expressions  are  polynomials,  we  ob- 
tain the  H.  C.  F.  by  factoring.     Hence 

THE  GENERAL  METHOD  FOR  FINDING  THE  H.C.F.   OF  POLYNOMIALS 

88.   Factor  the  given  expressions. 

The  product  of  the  lowest  powers  of  the  common  factors  is 
the  required  H.  Q.  F. 

Illustrations : 

Ex.  1.    Find  the  H.  C.  F.  of  a^-  3  a^- 10  a,  a^-  8  ^2+ 15  a, 
and  a^  —  25  a. 

a3  -  3  a2  _  10  a  =  a(a  -  5)(a  +  2) 

a3_25a  =  a(a  +  5)(a-5) 
H.  C.  F.  =  a(a  -  5).     Result. 


142  HIGHEST  COMMON  FACTOR 

Ex.  2.   Find  the  H.  C.  F.  of  a*  +  27  a,  2a^  +  a^-  15  a\ 

9  a  +  6  ^2  +  ^3,  and  a^  -  81  a. 

a^  +  27a  =  a(a  +  3)(a2  -  3  a  +  9) 
2  aS  +  a2  -  15  a  =  a(a  +  3)(2  a  -  5) 
9  a  +  6  «*^  +  a8  =  a(3  +  a)(3  +  a) 

a^-Sla  =  a(rt2+  9)(a  +  3)(a  -  3) 


H.C.F.  =  a(a  +  3).     Result. 

Exercise  51 
Find  the  H.  C.  F.  of : 

1.  ah  +  J,  ae  -{-  c.  11.    a^  —  ay^^  a^—2  c^y  +  a?/^^ 

2.  ax-\-hx^  ay-\-hy.       12.    Saa:^— 3a,  aa;2_  ;[o«2;  + 25a. 

3.  aa;  4-  a,  ax  — a.         13.   or^  —  5  a;^  —  14  a;,  2  a::^  +  4  a;^. 
.4.   m^  +  m,  m^  —  m.        14.   a^ -f  27  a,  a^  — 3a2  +  9«. 

5.  c^  +  c,  c^-1.  15.  a;2-3a:  +  2,  a:2_l^  x^  -  x. 

6.  a2_9^  a2_^3a.  16.  m^-9,  2m^-6m,  m^-m-6. 

7.  a5  +  ^  a%-\-ab.  17.  5m2-10mw+5w2,  lOm^-lOw^. 

8.  m2-l,  (m-l)2.  18.  a^  _  i^  ^3  _  ^^  ^4_i^  ^3  _  i. 

9.  rcHl,  (a:  +  l)2.  19.  1-a:^  1  +  5 a^  +  4 a^,  x-\-a^. 
10.  a^  — 1,  3^-\-x-\-l.  20.  a2_^^^^^^2;,  a*  +  a,  a*  +  a^. 

21.  2a2+5a-3,  3a2  +  5a_12. 

22.  a2  +  5a  +  6,  a3  4-"3«2+2a,  a2+7^  +  io. 

23.  a2-(^»-02^  ^_(a_5)2. 

24.  lO-lc  +  e^,  16-10(?+c2,  6-c-c2. 

25.  ar2+4a:-21,  3ar2-8a;-3,  3a^-14a;+15. 

26.  12a^-12b^  9a2_962,  13a^-15b\ 

27.  47w24-17w-15,  25  +  10m  +  m2,  3m3-75m. 

28.  a^-a^c-Sx^-\-Sex,  x^-\-ax^-^a^-Sax, 


i 


CHAPTER  XI 
FRACTIONS 

TRANSFORMATIONS 

89.  An  Algebraic  Fraction  is  an  indicated  quotient  of 
two  expressions. 

Thus,         2,    ^±*,    5i±l!,    «i±«*±*'  are  fractions. 
h      a  —  b     x^  —  y^  a  ■\-  h 

90.  The  Numerator  of  a  fraction  is  the  indicated  divi- 
dend; the  Denominator  of  a  fraction  is  the  indicated 
divisor. 

Thus,  in  the  fraction,  "^  "^  ^  ^ 

a^  —  2x 

a^  +  2  a;  is  the  numerator,  a^  —  2  a:  is  the  denominator. 

91.  An  Integral  Expression  is  an  expression  that  does 
not  contain  a  fraction. 

THE  SIGNS  OF  FRACTIONS 

92.  The  Sign  of  a  Fraction  is  independent  of  the  signs 
of  its  numerator  and  denominator. 

Thus,  +^,    +-,     H — ^  are  positive  fractions. 

, — , ^^-  are  negative  fractions. 

n  5  a:  —  1 

143 


144  FRACTIOJ^S 

The  sign  of  a  fraction  may  he  affected  hy  the  sign  of  its 
numerator  or  denominator  or  by  the  signs  of  both  together. 

Thus,     +.ll^=-«,    +ZlE=+«,    _Jz£=+«,    _ZL^=_« 
'6  5'-6&'  b  6'        -b         b 

Again,  the  signs  of  the  factors  in  the  numerator  and 
denominator  of  a  fraction  may  affect  the  sign  before  the 
fraction. 

Thus, 


a 


(1)  (+«)       .^+±,  (3) 


(+6)(+c)  be  ^'  (_6)(_c)        'be 

-«        =^^.  (2)  (-«)       =-±.  (4) 

From  which  we  make  the  following  important  con- 
clusions : 

From  (1)  and  (4): 

93.  An  odd  number  of  negative  factors  in  numerator  and 
denominator  together  may  he  changed  in  sign  provided  the 
sign  before  the  fraction  is  also  changed. 

From  (2)  and  (3): 

94.  An  even  number  of  negative  factors  in  numerator 
and  denominator  together  may  he  changed  in  sign  without 
affecting  the  sign  before  the  fraction. 

The  following  important  principle  underlies  many  pro- 
cesses with  fractions : 

95.  Multiplying  or  dividing  both  numerator  and  denomina- 
tor of  a  fraction  by  the  same  quantity  does  not  change  the 
value  of  the  fraction. 


BEDUCTION   TO  LOWEST   TERMS 


145 


REDUCTION  TO  LOWEST  TERMS 

96.  To  reduce  a  fraction  to  its  lowest  terms  is  to  change 
its  form  without  changing  its  value. 

A  fraction  is  in  its  lowest  terms  when  the  numerator  and 
denominator  have  no  common  factor . 

45  a%^c  ^3x3x5  a%^c  ^  3  ab^ 

4 


Ex.  1. 
Ex.  2. 


60  a%^c     4x3x5  aWc 

a^-\-  ax  _       a(a-{-x) 
a^  —  x^ 


Result. 


a  —  x 


Result. 


Result. 


{a -\- x^{a  —  x') 

Ex    3       a;2  +  5a:  +  4  ^{x^V){x-\-\^  ^x^-\ 
a:2  +  8a:  +  16      (ic+4)(a;  +  4)      a; +  4 

Hence,  the  general  principle  : 

97.   To  reduce  a  fraction  to  its  lowest  terms : 

Factor  both  numerator  and  denominator. 
Cancel  the  factors  common  to  both. 

Factors  common  to  the  numerator  and  denominator  may 
be  canceled. 

Terms  cannot  be  canceled  under  any  condition. 

Exercise  52 

Reduce  to  lowest  terms  : 


3. 


ia^ 
12  ab 

15  x^^ 

35  xY^ 

2Sa^f 


4. 


6. 


F.  H.  8.  FIRST  YEAR  ALG. 


39  m^n 
65  mn^ 

76  a^y^z 

23  a^c^ 
69a^cd' 
10 


8. 


9. 


48  m%y 

72  m%%  ■ 

135  a^mn^x^z^ 
81  a^nxh^ 

a^-\-  ax 
a^-{-ac 


c^  —  a 

ax-\-a 

bx^^l^x 

dx  +  2a 

4:X^-SX 

4:X^+20X 

a^  +  la^ 

a^-Sa^ 

a^  +  x^  +  x 

a^-1 

m^  +  Sm- 

-4 

m^—  5m  +  4: 

a3_^^2_6 

a 

a^-da 

2a^-la- 

-15 

2a^-lla 

+5 

2a^-2a^ 

146  FRACTIONS 

10.   r^.  15.   ^4+4=4^-     20.  «^  +  8«  +  T 

a*^  —  a^  +  5  a 

^^      ---     ■    ---.  16     ^^=^-  21. 

rr3  —  1 

12.  ^^^=: ;f^-  17.    — 22. 

X^  —  X 

-.«".•"  , «        a^  —  9  7^2 

13.  „   ^    „•  18. -•  23. 

(«  -  3  w)2 

14.    !^ ! 19. 24. 

x^  4-  a;  H-  1  m^  4-  w^2?^  4  a^  —  4  aj^ 

25     4tz3  +  5a2-21a  28    ^-(^-0' 

a*  +  2Ta  *    (3-c)2-a2 

26^    a2-(6  +  02  ^^^     2^  +  (^  +  ^'):r  +  a3 

(a  — 5)2— ^2  a;2_|_^^  _[_  5^^_j_^j 

27.    ^^= ■ ^'  30.    ■ • 

(a:  +  l)2_2;2  a;2_i4.2a-a2 


REDUCTION  TO  MIXED  EXPRESSIONS 

98.  A  Mixed  Expression  is  one  that  is  made  up  of  both 
integral  and  fractional  terms. 

1                         1 
Thus,  a  +  -  and  x  +  1 are  mixed  expressions. 

a  a;  +  1 

When  the  numerator  of  a  fraction  is  of  a  degree  higher 
than  the  degree  of  the  denominator,  it  is  possible  to  change 
the  fraction  to  a  mixed  expression  by  the  process  of  divi- 
sion.    Thus, 


REDUCTION  TO  MIXED  EXPRESSIONS  147 

Ex.  1.     Reduce  the  fraction 

^  "^      to  a  mixed  expression, 
a  —  1 


By  division  we  obtain : 


a3  +  1  (g-  1 

gs-  g^  g2  +  g  +  1 

g2  +  l 

g^  —  g 


g  +  1 
g-1 

+  2 

The  quotient  is  g^  +  g  +  1 ;  the  remainder,  +  2.  Hence  the 
remainder  is  the  numerator  of  a  fraction,  of  which  the  divisor, 
g  —  1,  is  the  denominator. 

Hence,  ^-±1  =  a'^^a  +  l^-  — ^.     Kesult. 

g  —  1  g  —  1 

From  which  we  state  the  general  principle : 
99.   To  reduce  a  fraction  to  a  mixed  expression ; 

Divide  the  numerator  by  the  denominator. 

Write  the  remainder  over  the  denominator^  and  annex  the 
resulting  fraction  to  the  quotient ;  remembering  that  the  sign 
of  the  remainder  becomes  the  sign  of  the  fraction. 


Exercise  53 

Reduce  to  mixed 

expressions : 

1. 

x-^l 

^'       a-\     ' 

2. 

a*  4-1 

x'-Sx-2 

g    2^-2x^-hx-\-l 
x^—x—1 

^    a:*  +  4  a;  +  1 
a  +  l  X  —  5  x^-\-  x  —  1 


148 

FB  ACTION  8 

^    s^  +  x^~2 

'       x^-hl 

x^  +  x^-2 

^'+1                           riS 

x^  +  x 

^+1      ^  +  i-^v4-  A^^s^i*- 

x^-x-2                                 -^    '   ^ 

x^          +1 

-x-S 

w2-4               '  a2^a-l'               •          x^  +  1 

REDUCTION  OF  MIXED  EXPRESSIONS 

By  Art.  99, 

-^+l=.a.l.        V. 

a  —  1  a  —  1 


Reversing  the  process, 


a  +  l+_2_^(«  +  l)(a-l)+2 
a  —  1  a  —  1 

^0^^-1  +  2 

=  ^^i±i.      Result. 
a-1 


Hence, 


100.   To  reduce  a  mixed   expression   to  an   improper 
fraction  ; 

Multiply  the  integral  expression  hy  the  denominator  of  the 
fraction  and  add  the  product  to  the  numerator^  remembering 


REDUCTION  OF  MIXED   EXPRESSIONS  149 

that  the   sign    of    the  fraction    becomes    the    sign    of    the 
numerator. 

Write  the  given  denominator  under  the  result. 


Exercise  54 

Reduce  to  improper  fractions  : 


1. 

a 

4. 

,    h 
ax  -\ — . 

X 

7.    ""-^x. 

X 

2. 

X 

5. 

1 

X . 

X 

8.  -+a, 
z 

3. 

c 

6. 

1 

a . 

a 

9.  ?-a. 
a 

10.   x-^l-\ .  \Z.  01^—  x+l-\- 


x-\-l  x-\-\ 

11.  <.  +  2+-5_.  14.  ^+2+^^?L±l). 

a  +  2  X  —  6 

x—^  4a2_j.2a4-l 

16.    -— ^  +  «  +  1. 
a  —  1 

Suggestion  :  Inclose  the  integral  expression  in  a  parenthesis. 

Then,  ^+l  +  a+l  =  t+l±Sl±M«^Z^ 

a  —  1  a  —  1 

^  "'^       Result. 


a-1 


150  FRACTIONS 

17.    --]-x+b.  19.    — — — =i— +  a  — a;  — 1. 

X  -\-  z  a  -\-  X 

18.  ?!^+4-:r2.  20.    <^~^)  +a:2  +  a;  +  l. 
X  +  z  x^  +  x—1 

21..  +  !-^^  +  ^. 

The  sign  before  the  fraction  becomes  the  sign  of  the  numerator. 

Hence,   .  +  1  _3^+j^  (^  +  DC^  +  2)  -  (3.  +  2) 

x  +  2  .r  +  2 

a;2+3:i;  +  2-3a:-2 


x  +  2 


=  — ^— .     Result. 


23.1-x  +  :fi--^.  25.  a2+2-^"-"'  +  "'. 

1  +  a;  a  —  1 

26.   a^  +  a  —  2  — 


a-1 


27.    x^-    x-4:-h 


x-\- 


i)- 


oo       2     To         fi      2-8wM 
28.  m'^  —    2  m  —    1 —  1  • 

29    2a      (^-l)(^+'^)  +  (^-B)      1 
a  +  1 


CHAPTER  XII 

LOWEST   COMMON   MULTIPLE.     LOWEST 
COMMON   DENOMINATOR 

THE  LOWEST  COMMON  MULTIPLE 

101.  A  Common  Multiple  of  two  or  more  algebraic 
expressions  is  an  expression  that  will  contain  each  of 
them  without  a  remainder. 

Thus :  a%^  will  contain  a%'^,  a%\  a'^b^  and  a%^ 
Hence,  a%^  is  a  multiple  of  a%%  a%^,  a^b^,  and  a^b^. 

102.  The  Lowest  Common  Multiple  of  two  or  more  alge- 
braic expressions  is  the  expression  of  lowest  degree  that 
will  contain  each  of  them  without  a  remainder. 


Thus: 


Given,  ■ 


a^b^  is  the  expression  of  lowest  degree  that  will 
contain   each   of    the    four   expressions   without   a 
„,   remainder. 


From  this  we  make  the  following  conclusion : 

103.  The  Lowest  Common  Multiple  of  two  or  more  expres- 
sions is  the  product  of  the  highest  powers  of  all  the  different 
factors  in  the  given  expressions. 

The  abbreviation  "L.  C.  M."  is  commonly  used  in 
practice. 

151 


152        COMMON  MULTIPLES  AND  DENOMINATORS 

THE  L.C.M.  OF  MONOMIALS 
Oral  Exercise 
Find  by  inspection  the  L.  C.  M.  of  : 

1.  a?h  and  aV^.  7.    IQmH  and  24^2^. 

2.  mV  and  mV.  8.    9  ah'^c^  and  12  a^5c. 

3.  2  m^a;  and  3  m^a;.  9.    10  (P'dx  and  8  ed?y. 

4.  3  a52  and  6  a^.  10.    15  a;^^  and  20  A. 

5.  8  mx  and  12  na;.  11.    35  a%h^  and  42  w^Jw. 

6.  lOx^y'^z  and  ^xy^^.        12.    51  am^  and  34  a/i^. 

THE  L.C.M.  OF  POLYNOMIALS 

In  finding  the  L.  C.  M.  of  polynomials,  we  must  first 
separate  the  expressions  into  their  factors,  as  in  the  case 
of  finding  the  H.  C.  F.     Hence, 

The  General  Method  for  finding  the  L.  C.  M.  of  Polynomials 

104.   Tractor  the  given  expressions. 

The  product  of  the  highest  powers  of  all  the  different 
factors  is  the  required  L.  Q.  M, 

Illustrations : 

Ex.  1.   Find  the  L.  C.  M.  of  a^  -  ah  and  ah  -  h\ 

a^  —  ah  =  a(a  —  6) 
a&-&2  =  6(a-6) 


L.  C.  M.  =  a&  (a  -  6) .     Result. 


THE  L.  C.  M.    OF  POLYNOMIALS  153 

Ex.2.   Find  the  L.  C.  Mo  of  x^  +  x-2,  a^-x-Q,  and 

x^  -{-  X  -2  =(x  -  l)(x  +  2) 

x^-x-Q  =  (x-{-2)(x-d) 

x^-4:X  +  S  =  (x-l)(x-S) 

L.C.M.=(a;-l)(a;  +  2)(a;-3).     Result. 

Ex.3.   Find  the  L.C.M.  of    a^ -\- 4.  a^  +  4  a,    aH-4:x, 

and  5  a2  -  20  a  +  20. 

a8  +  4  a2  +  4  a  =  a(a  +  2)  (a  +  2) 

a^x-^x  =  x(a-\-2){a-2) 

5  a2  -  20  a  +  20  =  5(a  -  2)  (a  -  2) 


L,C.M.  =  5ax(a  +  2)2(a-2)2.     Result. 

Exercise  55 
Find  the  L.  C.  M.  of : 

1.  a^  -\-  a^  ac  -\-  c.  9.   a^^  —  9,  a:^  —  6  a:  +  9. 

2.  ax  —  a^  x^  ~  X.  10.   rrfi-\-2m-\- 1,  m^  —  m. 

3.  m^  +  m,  m^  —  m,  11.   (P'  —\^  c^  —  1. 

4.  c2  +  2  c,  c2  -  2  (?.  12.   a^  -  4  a;  +  4,  2:3  _  8. 

5.  a:2_^^  (^2-1).  13.   4:a^-12a  +  9,  4^2-9. 

6.  a2  +  3a,  (a -3)2.  14.   a^-6a;2  +  9a:,  a;5_8l2;. 

7.  2<a:  +  X),  33:24.3^.  15.   a;2  +  3a;  +  2,  a^  +  4a;  +  3. 

8.  5a:2-15a:,  3a;3-27a:.         16.   x(x^- x -\-l'),  x^+1. 

17.  5a3_l25a,  2a3-20a2  +  50a. 

18.  (m  +  ^)^,  (w  —  ^)^  w^  —  ^^. 

19.  c\c-d},  c(ic^-d^),  c  +  d. 

20.  4(62  + 5m),  6(5m-m2),  8(^2  _;^2) 


154       COMMON  MULTIPLES  AND  DENOMINATORS 


21.  2m^  —  m^  —  m^  2  m^  —  S  m^  —  2  m. 

22.  Sa^-^x^  5a^-10x^+5x. 

23.  2a^-2a,  3  a(a  -  1)2,  4  a(a  + 1)2. 

24.  2^  +  1,  2  +  Sx  +  a^,  (a;+l)(l  +  a;). 

25.  a^-9,  a3-27,  a* -81. 

26.  ax  -{-  a  -\-  X  -{- 1^  ax^  -\-  ax  —  x^  —  x. 

27.  ax+Sa-2x-6,  a(a-2y,  a\a^- 


8). 


REDUCTION  TO  EQUIVALENT  FRACTIONS  WITH  A  COMMON 
DENOMINATOR 

105.  The  Lowest  Common  Denominator  of  two  or  more 
fractions  is  the  lowest  common  multiple  of  their  denomina- 
tors.    (Usually  abbreviated  thus,  "L.  C.  D.") 

The  process  of  reducing  fractions  to  their  lowest  com- 
mon denominator  is  illustrated  as  follows : 


Reduce 


5x    Sx 


,  and  -  to  equivalent  fractions  hav- 
4  6 


ing  a  common  denominator. 

The  L.  C.  M.  of  tlie  denominators,  3,  4,  and  6,  is  12. 
Hence,  12  is  the  lowest  common  denominator  of  the  three  denomi- 
nators, and  the  required  denominator  of  the  equivalent  fractions. 


12 


12 


4,  1st  quotient.        Hence,  — ;—  = 


4  x5a: 
4x3 

20  a: 
12 

1st  Resul 

3x3^ 
3x4 

_9x 
12' 

2d  Result. 

2  X  a: 
2x6 

_2x 
12* 

3d  Result. 

—  =  3,  2d  quotient.  Hence, 
4 

12 

—  =  2,  3d  quotient.  Hence, 

20  X     9  X  2  X 

Therefore,     ,  — ,  and   —  are  the  required  fractions  having 

12       12  12                     ^                                     ^ 

the  same  denominator,  each  equivalent,  respectively,  to  the  given 

fractions. 


nEDUCTtON  TO  EQUIVALENT  FRACTIONS        155 
2.    Reduce   -^^^ —  and  ^^^ —  to   equivalent  fractions 

X^  +  X  X^  —  X 

having  a  common  denominator. 

The  L.  C.  M.  of  the  denominators  x^  -\-  x  and  x^  —  xis,  x  (x^  —  1). 
Hence,  x  (x^  -  1)  is  the  L.  C.  D. 

^i^!:::ll  =  x-l,  1st  quotient.     (^-1)(^-1)      _felili.   Ist  Result. 

^^^-1)^,^1,  2d  quotient.      (^  +  1)  (^  +  1)       i^±iy     2d  Result. 

Therefore,   A^-^ — i—  and     ^^"^   ^ —  are  the  required  equivalent 
x(x^  —  1)  x(x^  —  1) 

fractions  having  the  same  denominator. 

From  these  processes  we  state  the  general  method. 

106.  To  reduce  two  or  more  fractions  to  equivalent 
fractions  having  a  common  denominator. 

If  necessary/,  reduce  the  fractions  to  their  lowest  terms. 

Find  the  lowest  common  multiple  of  the  given  denominators^ 
for  the  lowest  common  denominator. 

Divide  each  of  the  given  denominators  into  the  common 
denominator. 

Multiply  both  numerator  and  denominator  of  each  fraction 
by  the  respective  quotients  obtained. 

Exercise  56 

Reduce  to  equivalent  fractions  having  a  common  de- 
nominator : 


3^     2^  3    A  A  1 

4  '     3  '  '    ^x'  2x  X 

5a    2a  5  3  4 

6  '     9  *  '   2x'  4.x  3a;" 


156       COMMON  MULTIPLES  AND  DENOMINATORS 
5     2      2,^-  13.  '  ' 


ah     he    ac  '   xQc  —  T)'  a;(a;— 3) 

6.   A,   J_    A_.  14  ^  ^ 

^y''  x^y^^  xy^  '   2  a; +  3'  4a;2— 9 


a%^  a%^^  ah"^ 


a  b  c 

TT^nx    miP^'x    m^nx^ 

x-{-l     x  —  1 


a  +  6     a—  6 


11.  — ^,    ^ 19. 

^2_l'     (a;-  1)2 

2  2 

12.    — , -— .  20. 


a3- 

■1'  a3  +  a2  +  a 

3 

1 

a^- 

■x-Q' 

a^2+2a;- 

15 

1 

2 

2x^ 

-a^-1 

.'  Qx^-x- 

-2 

2 

3 

a*  4- a 

a2- 

■a+1' 

X 

a;2  — re— 6' 

1 

x^j^^x+2' 

X 

1 

2              3 

aQa  +  iy   (a  +  iy  '   (a-\-hy(a-i-hy\a+hy 

X  X  X  .  • 


21. 


22. 


23. 


2(0^4-1)'  3(a;-iy  2(rr2-l)  . 
2x  4x  Sx 


5(a:  + ly  3(a;-iy  2(a:2_2a;  +  l) 

X  X  X 

a^  —  x-^-V  x^  -\-x    x'^  +  x 
X        a^  +  x^  -\-x        X 


25. 


1  1  i 


8^4-15'  5w8-125w'  7w-35'  6m4-30 


CHAPTER   XIII 
FRACTIONS   (Continued).     REVIEW 

I.    ADDITION  AND  SUBTRACTION 

If  two  or  more  fractions  have  the  same  denominators, 
their  numerators  may  be  added  and  a  single  fraction  will 

result. 

„,  2,3     5 

Thus :  =  +  =  =  =• 

7     7     7 

a      b      c      a  -\-  b  +  c 

XXX  X 

X      y      z  _x  —  y  —  z 
a      a'    a  a 

By  Art.  106  we  may  reduce  any  given  fractions  to 
equivalent  fractions  having  a  common  denominator ;  hence 
it  follows  that  any  given  fractions  may  be  added. 

The  following  illustrations  will  serve  to  show  the 
process : 

Ex.  1.     Simplifv  -^+-^- 

The  L.  C.  D.  is  a^  -  b^ 

Dividing  each  denominator  into  the  L.  C.  D.  and  multiplying  the 
corresponding  numerators  by  the  quotients,  we  obtain 

a       .       b     _a(a  +  b)  +  b(a  —  b) 


b     a  +  b  a^-b'^ 

_a^-\-db-\-db-b'^ 
a'^-b'^ 

_ai^2ab-b'^ 
a^-b^ 
157 


Result. 


158  FRACTIONS 

Ex.  2.     Simplify       -^ -   ,    ^     ,  +  ^f~^ • 

TheL.C.D.  isa:8+l. 

Dividing  each  denominator  into  the  L.  C.  D.,  multiplying  the  cor- 
responding numerators  by  the  respective  quotients,  and  writing  the 
result  over  the  L.  C.  D.,  we  obtain 

X  X  .  2x^-1  _x(x^-x  +  1)  -  x(x-\-l)  +  2x^-l 


4- 


x+lx^~x+l       x^  +  l  x^+1 


x^  -x^  +  X  -  x^  -  x-^2x^-l 

X3+  1 

x^-l 


a;8+  1 


Result. 


From  which  we  make  the  statement  for  the  general 
process : 

107.   To  add  algebraic  fractions : 

Reduce  the  given  fractions  to  equivalent  fractions  having  a 
common  denominator. 

Write  the  sum  of  the  numerators  of  these  fractions  over 
the  common  denominator.,  remembering  that  the  sign  of  each 
fraction  becomes  the  sign  of  its  numerator  in  the  addition. 


Simplify : 


1. 

«  +  *. 

X      x 

2. 

c      Ze 

a^2a 

3. 

xy     xz 

Exercise  57 

4. 

A^A. 

ax      bx 

5. 

\*h\- 

6. 

1      1       1 
m     71     p 

7. 

2     2     3^ 

a     X     y 
2       2       4 

8. 

ah      ac     he 

9. 

5         3        2 

mn     mp     np 

10. 

111 

xij     xz     yz 

ADDITION  AND  SUBTRACTION  159 


S  ah     b  ah     ah 
'      4        ~6~'^T 


14. 


,_     a  —  '^  +  x.  X  —  a-\-\ 
J-5'   7i 1 : 


12.  ^+i+^=i. 


^^'   ~^^^2" 
x-Z     x-1 


14 


5x  —  5a-\-c     c-\-Sa-\-Sx 


17. 


5  3 

4a-3c  +  l      Sa-\-c-l 


18.    a+2      g-l  ^  g  +  l 


19. 


6.  3  2 

2a-l      7a+3a-3 
5  10  2 

a;-  5      a;+  3      2a:-  5 
12  6  9      ' 


21.    1+-1-.  24.       ^ 


aa  +  (j  7m3m  —  1 

oo      ^  J      2  o«     2  1 

22. 1 -.  25.    — -  — 


2a;a:4-l  ah      a  —  h 

23.    -.,^  +  i.  26.    -1^+      1 


a^— 4a  a;-f-la:— 1 


160  FBACTIONS 


27.  -^ ?-.  36.  ^±1  +  2^. 

m  —  n     m  -\-  n  a  -{-  2     a  -{-  S 

28.  -^ ?-.  37.  ^±l  +  ^-Zl. 

29.  ^ 2^.  38.         "+1    .+       «-l 


a-2      a  +  3  a2  +  a  +  l      a^-a+l 

5             5                          „^     a  —  2  a;  ,       4  «a; 
30. .  39. h 


^^2  a-3  6 

31.    — -H -— .  40. 


a(^a  +  1)      a(a  -  1)  a  +  3      a^  -  9 

32.  — ^ ?__  ^^     ^  +  I_£^. 

a(a  -  1)      a(a  +  1)  ic  -  7      a;  +  7 

33.  J—^-J-,  42.     .^-A.  +  ^  +  ' 


^2  _^  ^  '  ^2  _  ^  (a:  +  1)2      (a;  -  1)^ 

34    _JL__l£_  43     3a;  +  2      32;-2 

•    l_^      l_a;2-  •    32^-2^3:^:4-2* 

Sx         2x  ^1^7^  +  4     7w-4 

35. —»  44.     — 7  —  — J. 

rr-1     x-\-l  7w-4      7m  +  4 

w  —  2 m  +  2 

*   ^2— 2m  +  4     m2  +  2w  +  4* 

46.     ,  .A  .,+  ^ 


47. 


(^  +  l)(^+2)      (a;  +  l)(a;-2) 

5 2_ 

(a:-l)(2:  +  l)     cx-iy 


MULTIPLICATION  AND  DIVISION  161 

4  2 


48. 


49. 


a2+3a  +  2     a^  +  Ba  +  Q 
1  1 


2^-{-lx-{-12     x^-}-Sx+15 


ah           ah     ,        1 
50. -f- 


a-{-h      a  —  h      a^  —  h' 
51.        2^-^^+      ^ 


52. 


^2  —  1      m-\-l     m  —  1 
a-2      a  +  2     a2-4* 


53.    .        ^    ■       +      1  1 


54.        „       1         ,-^i-+        1 


m^— w  +  l      m*  +  m      w  +  1 
55.    .JU+^J-.+         1 


a;2  —  a^      {x  +  a)2      (x  —  a)^ 

56.    ^Il1-^^iJ+^(^^-^), 
x-{-%       X  —  4:       X^  —  X  —  12 


II.   MULTIPLICATION  AWD  DIVISION 

The  processes  of  multiplication  and  division  of  algebraic 
fractions  are  not  unlike  the  same  processes  with  arithmeti- 
cal fractions. 

(1)  Multiplication. 

The  product  of  two  or  more  fractions  is  the  product  of 
the  numerators  divided  by  the  product  of  the  denominators. 

F.  H.  S.  FIRST  YEAR  ALG.  — 11 


162  FRACTIONS 


Thus:  «x-^x^"  =  ^. 

X     y     z     xyz 

ah      cm     nx  _  abcmnx 
mn      bx     ay      mnhxay 

_  c 

~  y 


Canceling, 

(2)  Division.  ^ 

The  quotient  of  two  fractions  is  obtained  by  inverting 
the  divisor  and  using  it  as  a  multiplier,  whence  the  process 
becomes  a  multiplication  of  the  given  fractions. 

Thus: 


a  .  d  _a      h  _ah 
c      b      c      d      cd 

am 

bm  _  am      xz  __az 

xy 

xz       xy     bm     by 

The  general  process  may  be  stated  as  follows  : 

108.  To  multiply  a  fraction  by  a  fraction  : 

Multiply  the  numerators  together  for  a  new  numerator^  and 
the  denominators  for  a  new  denominator. 

By  cancellation  reduce  the  resulting  fraction  to  its  lowest 
terms. 

109.  To  divide  a  fraction  by  a  fraction  : 
Invert  the  divisor  and  proceed  as  in  multiplication. 
The  following  examples  will  illustrate  the  processes  : 

Ex.  1.     Multiply  1^^  by  i^. 
^^    %mny    ^  Z  ax^ 

\^€?-x     4  my  _  15  X  4  a^xmy 
8mny      dax^     8  x  dmnyax^ 

2nx 


I 


MULTIPLICATION  AND  DIVISION  163 

Ex.  2.    Multiply : 

x-y      ix-yy 
Factoring  numerator  and  denominator  in  the  multiplier,  we  have 

x  +  y  ^  ^!_z_ii  =  ^±1 X  (^  +  y)(^  -  y) 

x-y      {x  -yy      x-y      {x  -  y){x  -  y) 


Canceling, 


Ex.  3.    Simplify : 


i^±y^.   Result. 
{x  -  yy 


Factoring  the  terms  of  the  second  and  third  fractions,  we  have 

g  +  l^^     (a-\y     .  a^-l  ^a+1  ^^  (a-l){a-l)  .   (a  +  l)(a-l) 
a-l      a2-3a  +  2  "  a^-^      a-l      (a-l)(a-2)  '  (a  +  2)(a-2) 

Inverting  the  divisor,       =  ^^  x  (^"^X^"^)  x  («  +  2)(«-2) 
^  '  a-l      (a-l)(a-2)      (a  +  l)(a-l) 

Canceling,  =  ^^-^.    Result, 

a  — 1 


Simplify : 
aJc      6  X 

^     8  ca;      10  m2 
0  m         y  a; 

•    9i/       2ac 


Exercise  58 


,     21^25      5a^ 
10  mx     8  «S 

89  a(?2      5  moi^ 

6  m^    ^  2  m^ 
*   13^  '  39^' 


164  FRACTIONS 

^    16  ahc  .  32  ao  ^^      rn^-l       2m^+4:nv' 

'    27  mx  '     9x' 

8    1^_21^  ^^ 

32  m/i     40  WW 


3£_  4w^  25^ 
5  m  15  a;^  12  w 


15. 


35  g3^   18  aH^x     28  y 
'   12  a^«/  '    49  c^    '  45  a^Jc'       * 

or  —  X     x^  +  x 


m2+  2m 

m^-\-m 

2^2-1 

2^3  +  1 

^-1 

C?-l 

^+1    *    C2 

-c+1 

a  +  1  .  a2 

-1 

a  -  2  '  a2 

-4 

:?;  -1 

:i:2_l 

2^3  +  1   •  :^2 

-a;+l 

a^—xy     x^  —  4:xy+^y'^ 

a^-9       5a;  +  10  i^-^y  x^-y' 

X  X         x^—1 


19. 


20. 


21. 


22. 


23. 


24. 


25. 


a;  +  1     x—1        ax 

2(a-{- 1)2     5  (g  -  1)3  ,        5  a 
3(a-l)2  '  6(a-l)2  *  8(a+l) 

a2^>2g2  _  4  ^252g2_^5g_6 

a^^(P-9  '  a%'^(^  +  ^  ahc  +  4^  ■ 
a2_3^_18  a3_27 

(a:  +  l)(a;+2)     (a;+3)(a:+5) 
(aj  +  3)(a;  +  l)     (a:+2)(a:+7)* 

a^-a:-2        rc2-3a;-4 
a^_5^+6  *  a;2_73,+  i2' 

6-\-  x  —  a^     4  —  4  a;+  a^ 
4-2^2      '9_eJa:  +  ar2" 


26. 


27. 


28. 


29. 


30. 


MISCELLANEOUS  FBACTIOKS  165 

^2-1  a2-2a-3 


a2  -  5  a  +  6 

'  a?- 

-3« 

+  2 

a;3  - 16  a; 

2;2 

+  3:r-4 

aa:^  —  7  a:z:  +  12  a 

'  x^ 

-4a;  +  3 

i?;3+l 

ax^- 

-  a 

x^  +  2x+l 

a^- 

-1 

4a2_l 

9a2 

-4 

6a^+a- 

-2 

6  a2  -  a  -^  2 

8^3 

+  1 

•  4a2_2a 

+  1 

«2  _^  2  a5  4-  ^ 

-^ 

a2- 

-2atf  +  ^- 

-62 

a2_2a6  +  6 

-(?2 

(«  +  ^  +  0' 

III.    MISCELLANEOUS  FRACTIONS 

It  frequently  happens  that  an  example  in  fractions  in- 
cludes addition  or  subtraction  with  multiplication  and 
division.  The  student  should  note  carefully  the  indicated 
order  in  such  problems,  remembering  that  parentheses 
indicating  addition  must  be  simplified  before  further 
operations  are  possible. 

Exercise  59 

Simplify : 

^  (i-i)(i-f}     '■  e-)e-> 


166  FRACTIONS 


•■'J 


Va;      a/\a;      ay  \a     b     eJ\ab  +  oc+ac. 


11 


12 


/^i  _  A^ -L- /'^  _  ^^  17     /^  — ^     m-{-n\f  m-\-n  \ 

W     f)\x     'yj  '    V    2  3    Adm-nJ 

/^_^y___a^___\    ^g^    /a  +  1     x-l\  ,  fx  ^  ^ 
\a      x)  \^  —  2  ax  -\-  d^j  \    a  x   J     \a       J 

\x-]-a       J\x  —  a       J     \Qo^—ay 

V     c     ^    )\^a-\-xy-cJ 


20. 


21. 


(x-^vf-(x-iy 


x-l  (^-1) 

'    \x      aj\        xj     \        x) 

-  (--:i7)(lf?l)(f-i-'> 

V.       1-xyJ     \         1-xy  J 


GENERAL   REVIEW  167 

GENERAL  REVIEW  — GROUP  IV 

THE  H.C.F. 

The  H.  0.  F.  of  two  or  more  expressions  is  made  up  of 
those  factors  that  are  common  to  the  expressions. 

The  prodvct  of  the  lowest  powers  of  the  factors  common  to 
the  given  expressions  is  the  H.  C.  F. 

THE  L.C.M. 

The  L.  C.  M.  of  two  or  more  expressions  is  made  up  of 
all  the  different  factors  that  occur  in  the  expressions. 

The  product  of  the  highest  powers  of  all  the  different 
factors  occurring  in  the  given  expressions  is  the  L.  C.  M. 

An  H.  0.  F.  divides.     An  L.  C.  M.  contains. 

FRACTIONS 

Signs 

The  sign  of  or  before,  a  fraction  is  independent  of  the 
signs  of  the  numerator  and  the  denominator  of  the  fraction. 

The  Signs  of  thk  Factors  of  the  Numerator  and  the 
Denominator  of  a  Fraction 

Changing  the  signs  of  an  even  number  of  factors  in  both 
numerator  and  denominator  of  a  fraction  does  not  change  the 
sign  of  tlie  fraction. 

Changing  the  signs  of  an  odd  number  of  factors  in  both 
numerator  and  denominator  of  a  fraction  changes  the  sign 
of  the  fraction. 


168  FRACTIONS 

Introducing  a  Factor  in  Both  Terms  of  a  Fraction 

Both  numerator  and  denominator  of  a  fraction  may  he 
multiplied  by  the  same  quantity  without  affecting  the  value  of 
the  fraction. 

Removing  a  Factor  from  Both  Terms  of  a  Fraction 

Both  numerator  and  denominator  of  a  fraction  may  he 
divided  hy  the  same  quantity  without  affecting  the  value  of 
the  fraction. 

The  Transformations  of  Fractions 

I.  To  reduce  a  fraction  to  its  lowest  terms  is  to  change  its 
form  without  changing  its  value. 

II.  To  reduce  an  improper  fraction  to  a  mixed  expression 
is  to  change  to  a  form  in  which  a  part  of  the  terms  are 
integral;  the  remainder^  fractional. 

III.  To  reduce  a  mixed  expression  to  a  fraction  is  to 
change  its  form  so  that  no  integral  terms  remain. 

IV.  To  reduce  a  given  number  of  fractions  to  equivalent 
fractions  having  a  common  denominator  is  to  change  the 
form  of  each  so  that  the  denominators  of  the  resulting  frac- 
tions shall  be  the  same.  The  operation  depends  upon  the 
introducing  of  the  same  factor  in  both  numerator  and 
denominator. 

Addition  and  Subtraction  of  Fractions 

Fractions  having  the  same  denominators  may  he  added  hy 
adding  their  numerators. 

Fractions  having  different  denominators  may  he  added 
after  they  are  changed  to  equivalent  fractions  having  a 
common  denominator. 


GENERAL   REVIEW  169 

Signs 

In  addition  and  subtraction  of  fractions^  the  sign  of 
each  fraction  becomes  the  sign  of  its  numerator  when  the 
numerators  are  added. 

Multiplication  and  Division  of  Fractions 

The  product  of  two  or  more  fractions  is  the  product  of  the 
numerators  divided  by  the  product  of  the  denominators. 

Cancellation^  or  the  removing  of  the  same  factor  from  both 
numerator  and  denominator  of  a  fraction^  reduces  the  result 
to  its  simplest  form. 

The  quotient  of  two  fractions  is  obtained  by  inverting 
the  divisor  and  using  it  as  a  multiplier.,  whence  the  process 
becomes  the  same  as  in  the  multiplication  of  fractions. 

Bzercise  60 

REVIEW 

Find  the  H.  C.  F.  of : 

1.  a^  +  x^  x^  —  X,  x(x  +  1)2. 

2.  a3_2a2  +  a,  (a-iy. 

3.  a2-l,  a^-1,  a^-1. 

4.  a^+a-6,  2a2-5a  +  2,  3a2-4a-4. 

Find  the  L.  C.  M.  of  : 

5.  ^2-6^-42,  2a2+6a-36. 

6.  (2:2-1),  (^-1)2,  x^-\-2x-^l. 

7.  a^-{-x^y^  xy^  —  y^,  x'y^  —  7^y^. 

8.  a2-l,  a2-2a+l,  a3-3a2+3«-l. 


170  FRACTIONS 

Find  the  H.  C.  F.  and  the  L.  C.  M.  of  : 

9.    c^-\-8cd^  2(^-Sc(P,  c^-i(^d  +  4:(^€p, 

10.  2a;2_2,  4ic3-4,  2a;2  +  2ic-4. 

11.  x^-{-x-2,  a^+2x-S,  x^-\-Sx-4:. 

12.  (a-\-bf-c^  P-(a  +  cy,  (h  +  cy-a\ 

Reduce  to  lowest  terms  : 

13.     5^- 20a;  ^^        125-27^:3 


lbx{x-2)^  25-30ic4-9a;2 

12  a -12  X  ,„  a:* -2* 

14-    ^ — 7. ^ \ — ^-  18. 


^^      g2  -  9  g  +  20  a:2-(a-l)2 

c2_7^^12'  *    (x  +  iy-a^' 

-  g      15  a:^  4-  a:^  —  6  a;  x^ -\- x -\-  ax -\-  a 

21  a:;^  —  a;^  —  10  a?  a;^  +  a;  —  ca;  —  c 

Change  to  mixed  expressions : 

a3_2a2_5  a:3_^  +  7 

21. 23.    — ' 

«2-l  a;2  +  a:+l 

a:^  +  a:2  -  7  2:g^-2^2_^_3 

'        x^+1     '  '            x^-1 

Change  to  improper  fractions : 

25.  ^  +  1+^.  27.    (:,  +  2)'-^(^-|^X 

a:—  1  a:  +  4 

26.  «2_a  +  i_«'(«-l).  28.    ^-a-3. 

a+1  a— 2 


GENERAL  BEVIEW  171 


29.    ^+2:2  +  2 


30 


Find  the  sum  of  : 
2      .  1 


33. 


34. 


2:^—1      x^  +  x  +  1      x  —  l 
1  1  3a 


(a  -  1)2        (a  +  1)2        (^2  _  1)2 


35.    -  + 


36. 


37. 


^_2  '  ic-1      (a;-2)(a;-l) 

2.2  4 


-^^x-i-2x^-\-^x+S      2:2  +  52:4-6 
3  2 


(2;-l)(2:-2)(2:  +  2)      (x  -  2) {x -{- 2) (x -\- 1) 

38.    ^ ^- + 

(a-6)(a-c)      {a-b)(h-c)      {a  —  c^(h-c) 

Simplify : 

^2_l  ^_\ 


39. 


ecg-2  6?2      g2_g^       c2_  3 g^ 4.2^^2 
•      e2  +  tf(^     *  (^-^)^  '  c2_^2         * 

2 


42 


^-)a->)-G-^^>Xra-'> 


172  FRACTIONS 


43. 


a2  +  2a-3     a^-Ga+S  '  (a-l)(3  +  a) 


g^  —  2  am  +  m^  —  x^     a  +  m  —  x 
cfi-\-2  am  -\-m^  —  a^     a  —  m  +  x 


45. 


/g  +  l  _  a-l\  ,  (a  +  1  .  a  —  l\ 


46.  Find     the    H.  C.  F.    of    m^  +  ac  -\-  am  +  cm    and 
m^  —  ac  -{-  am—  em, 

47.  Add  ^-^  I  ^  +  <^     m^-a^ 

a  m  4:  am 

48.  Reduce , r,    -— 5,    to   equivalent 

1-y     1-f     !  +  «/  +  / 

fractions  having  a  common  denominator. 

49.  Collect  —4 ^^4-       ^ 


l-2a     l  +  2a     1-4^2 

50.   Simplify  — — -  •  — — 7  X  - 
^     -^   ab  +  l     ab-1 


51.  Change         ~        "^      to  a  mixed  expression. 

52.  Simplify  ^-^  .  -.-^^  .  -— ^. 

53.  Find  the  L.C.M.  of  6a  +  2,  27a2-  3,  and  108 a^  +  4. 

54.  Find  the  sum  of 


x^-^Sx-10     x^-\'4tx-6 

Jhange    x^—2z 
fraction 


55.   Change    a^^- 2  a:- 1  +  ^^i^    to    an    improper 


56.    Collect 


GENERAL  REVIEW  173 

a  b 


(l^aXa  +  b)      (l-5)(a+5) 
57.   Find    the    L.  C.  M.  of  a^  +  Sa^-\-lQ,  a^-lG,  and 


58. 


59. 


Simplify  (.-l  +  -A3).(.-3--A^) 


60.   Find  the  H.  C.  F.  and  the  L.  C.  M.  of 

x^ -\- ax -{■  a  +  X,  x^  —  hx  +  X  —  h,  and  a^ -\-  ex -{- x  +  c. 


61. 


^^"p^^^^  (  <--9d^  H-^T^rr 


62.    Collect  -  + 


x+2     a^+S     x^-2x-{-4t 

2 


63 


Simplify  (^__f\^^^ 
^     ^   \x-l      x-lj        x-1 

64.  Simplify  fab +  5 +  ^]-^ fab -^4  +  ^' 

2 

65.  Change   a^  —  a^  +  a  —  1  -\ to    an    improper 

fraction.  ^ 

r.r>  \AA      ^-"^  X-2     .  12 

66.  Add 


x-^2     x-\-4:     x^-\-6x-\-S 

^'      I'f     ^^  +  2  am        a^—Sm^         a^-\-  4: am^ 
a^  +  4  w^     2  a%  H-  4  am^       a  — 2  m 

68.    Simplify  (^+2^)-(4^-2^> 


CHAPTER   XIV 

SIMPLE  FRACTIONAL  EQUATIONS.    PROBLEMS 

110.  To  Clear  an  Equation  of  Fractions  is  to  change  its 
form  so  that  the  fractions  disappear. 

Thus,  in  the  fractional  equation, 

we  may  multiply  both  members  by  5,  whence  we  have 

^=15. 
5 

Or,  reducing,  x  =  15. 

Wlien  two  or  more  fractions  occur  in  an  equation,  the 
process  of  clearing  is  similar  to  that  above,  but  the  multi- 
plier is  the  L.  Q.  M.  of  the  given  denominators. 

Thus,  to  solve  the  fractional  equation,  — —  ;;  =  s» 

4       0      0 


L.  C.  D.  =  12. 

3a;      a:      5 
4       3~6* 

Multiplying  by  12, 

9a:-4x  =  10. 

5a:  =  10. 

a:  =  2. 

Hence,  the   general   statement   for   solving   fractional 
equations. 

174 


SIMPLE  FRACTIONAL   EQUATIONS  175 

111.   To  solve  an  equation  containing  fractions  : 

Multiply  both  members  of  the  equation  by  the  L.  C.  D.  of 
all  the  fractions,  remembering  that  the  sign  of  each  fraction 
becomes  the  sign  of  its  numerator.  Complete  the  solution  by 
the  methods  already  learned. 

The  following  solutions  will  illustrate  the  general  prac- 
tice. The  student  should  carefully  work  out  each  step 
before  undertaking  the  exercises. 

Ex.1.     Solve^+?^  =  5. 
4  3 

The  L.  C.  D.  is  12.     Multiplying  both  members  by  12,  we  have 

3(a;+  l)  +  4(2a;-5)  =60. 

3a;  +  3  + 8a; -20  =  60. 

lla:  =  77. 

a:  =  7.    Result. 

Ex.2.     Solve  £^-^£±1=^.       - 

O  D  Z 

The  L.  C.  D.  is  30.     Multiplying  both  members  by  30,  we  have 
10(a:  -  1)  -  6(2  a;  +  1)  =  15(a;  +  1). 
lOx  -  10  -  12a;  -  6  =  15a;  +  15. 
-17  a;  =  31. 

a;  =  -  f f     Result. 


176  FRACTIONAL  EQUATIONS.      PROBLEMS 

Ex.3.     Solve  2^-^  =  2       ^"^ 


X-\-\       X  —  1  x^—1 

The  L.  C.  D.  is  x^  —  1.     Multiplying  both  members  by  x^  —  1,  we 
have 

(x-l)(x-  l)-(x  +  l)(x  +  2)  =  2(x'^-  l)-2a:2. 

(a:2  _  2  a:  +  1)  -  (:r2  +  3  a:  +  2)  =  (2  a;2  -  2)  -  2  x2. 

x^-2x+l-  x^-Sx  -2  =  2x^-  2-2a:2. 

~-5x  =  -l. 

—  \.    Result. 

Exercise  61 


2^     45_^^29 
*    3        5       2~15' 

8    ^+1 , x-1^2 

'      8  2         3 

a;  4-1      2a:-l_o 

10.  ^  +  ^  +  4  =  0. 

a;  +  l      a^-l_o 
^^*  "5  3 ^• 

2a:-8      8a;  +  l     x-1 


Solve : 

1. 

i^ 

2x 
3 

7 
=  — • 

2 

2. 

1+ 

2^: 
5 

11 
15* 

3. 

2a; 
3 

2 

5 
6* 

4. 

a;  — 

8a: 

2 

1 

2* 

5. 

1+ 

2a; 
5 

+  ??  = 
^15 

=  0. 

6. 

I* 

8a; 
2 

+f- 

25 
12* 

12. 


4  2  6 


,^     4a;-8  ,  8a;  +  l      ^-10 
13. ! —  = . 

2  7  14 

5a;-8      6a;-l      8a;-2^Q 
2  3  4* 


SIMPLE  FRACTIONAL   EQUATIONS  177 

15.  1(^  +  1) -1(^-1)  =2. 

16.  |(2:.-l)-f(3:.+  5)  =  J5. 

18.    Q^x}Q-x)  +  x^  =  0. 

x-i-2      5-xx-^lS 


19 


8 


Sx-^2      l-2x     ^-6^Q 
5  3  2  '     . 

21.  ^  +  3  =  ^. 

22.  i(:.-2)-i(^+2)-i(^-l)=0. 
4  a;  -  3      a^  4- 10 


23.   4 


2 


24.    Ax         ^^         "2-2 

o«     (^  +  1)^      Ca:  + 2)2  _  2-^2 

23. • 

3  2  6 

x  +  \      x—1      2a;— 1      3a: 


'''       3 

X 

-1 

a;-3 

X 

+  2 

a:  +  5 

X 

-6 

a:-7 

X 

+  3 

a;  +  5 

X 

+  1 

a:4-2 

X 

-3 

X-Q 

2 

a:-3 

2x-{-5 

2  4  6 

„„^— J.      u.  —  ^  „,     3a;— 1      2a;  —  1      -, 

27.    :;  = -'  31. =  1. 

a;  +  1         a;  —  1 

28.    = 32. — —  =  2. 

a;—  1      a:—  2 

23    ^_^^^^_^^  33     5a;-l_3^2a:  +  l. 

a;+2  a;-l 

30.    -----^---r-.  34    4a;-5_5a;-l      g^Q 

3a;  +  4      3a;+7  2a;-l       a;  +  l 

r.  H.  8.  FIRST  YEAR  ALG.  —  12 


178  FRACTIONAL   EQUATIONS.      PROBLEMS 

,,    X     x^-Sx     4  ^^    x+S     x-S         7 

35.     — —  =  — •  36. 


5      &x-^l      5  x-S     x+S     x^-d 

^„    2x-5     2x-\-5  8 

37. 


38. 


2x  +  5     2x-5     4tx^-2b 

x+2      a;-4^12 

X—  2     x  +  4i      X 

_    1/5^+3      2£-ll      a;  +  2 

^^*  2i""i       3~r-6- 

3.2  1 


40. 


41. 


42. 


2(a;  +  l)      22^-1      3(a:+l) 

Ifx      o\      2/:r      A    .10+£ 

^j^x'^-^      a^-x^  +  2 


x-\-l  x  —  1 

a;+l  x  +  2 

Sx  2x  X  5        2x 

45. 


46. 


x+1      3(:r  +  l)      2(a;+l)      6      x-i-l 

2      ^      3     ^     15 
a:—  3      2^+1      3a;— 1 


^„     a;2+2:  +  l,       X  x^—x  +  l 

47. h  — 


48. 


a:  +  l  a^-1  x-1 

X  a^-\-l  X 


2a;+2      ^^-^      %x-Z 


SIMPLE  FRACTIONAL   EQUATIONS  179 

PROBLEMS  PRODUCING  SIMPLE  FRACTIONAL  EQUATIONS 

In  our  first  application  of  the  equation  to  written  prob- 
lems we  made  only  such  statements  as  would  give  equa- 
tions without  fractions.  In  the  following  exercise  we 
shall  be  able  to  make  use  of  fractional  expressions  in  our 
.statements,  and,  consequently,  in  our  equations  as  well. 
The  method  by  which  we  reach  our  solution  will  differ  in 
no  way  from  the  method  employed  in  our  earlier  equa- 
tions. The  suggestions  for  solving  a  problem  (Art.  63) 
are  repeated  as  an  aid  to  the  solutions  required. 

In  solving  a  problem  : 

1.  Study  the  problem  to  find  that  numher  whose  value 
is  required. 

2.  Represent  this  unknown  number  or  quantity  by  x. 

3.  The  problem  will  state  certain  existing  conditions  or 
relations.     Express  those  conditions  in  terms  of  x. 

4.  Some  statement  in  the  problem  ivill  furnish  a  verbal 
equation.  Express  this  equation  algebraically  by  means  of 
your  own  written  statements. 

No  statements  or  solutions  are  given  in  the  following 
exercise.  With  the  aid  of  a  suggestion  whenever  a  new 
element  is  introduced,  the  student  should  make  his  own 
statement  with  little  or  no  difficulty. 

Exercise  62 

1.  The  sum  of  the  fourth  and  the  fifth  parts  of  a  number 
is  9.     Find  the  number. 

2.  Find  that  number,  the  sum  of  whose  third  and 
fourth  parts  is  5  less  than  the  number  itself. 


180  FRACTIONAL  EQUATIONS.      PROBLEMS 

3.  The  difference  between  the  fourth  and  the  ninth 
parts  of  a  certain  number  is  2  more  than  one  twelfth  of 
the  number.     What  is  the  number  ? 

4.  There  are  three  consecutive  numbers  such  that  when 
the  least  is  divided  by  4,  the  next  by  3,  and  the  greatest 
by  2,  the  sum  of  the  quotients  equals  the  largest  number. 
Find  the  numbers. 

5.  The  sum  of  two  numbers  is  21,  but  if  the  greater 
number  is  divided  by  the  smaller  number,  the  quotient  is  6. 
What  are  the  numbers  ? 

6.  The  difference  between  two  numbers  is  14,  and  when 
the  greater  number  is  divided  by  the  smaller  number  the 
quotient  is  2  and  the  remainder  is  3.     Find  the  numbers. 

(Hint  :  The  dividend  minus  the  remainder  will  exactly  contain 
the  quotient.) 

7.  A  man  sold  10  acres  more  than  one  third  of  his 
wood  lot  and  had  left  12  acres.  How  many  acres  were 
there  originally  in  the  wood  lot  ? 

8.  Two  numbers  differ  by  7  and  one  of  them  is  five 
fourths  of  the  other.     What  are  the  numbers? 

9.  The  denominator  of  a  certain  fraction  is  greater  by 
2  than  the  numerator.  If  1  is  added  to  both  numerator 
and  denominator,  the  fraction  becomes  f.  What  was  the 
original  fraction  ? 

10.  A  man  sold  a  cow  for  $  25  more  than  one  third  of 
what  she  cost  him,  making  $  5  by  the  sale.  What  was  the 
original  cost  of  the  cow  ? 


SIMPLE  FRACTIONAL  EQUATIONS  181 

11.  Out  of  a  certain  sum  a  man  paid  a  bill  of  f  45, 
loaned  one  fifth  of  the  remainder,  and  found  that  he  had 
remaining  f  44.     How  much  had  he  originally  ? 

12.  Find  two  consecutive  numbers  such  that  ^  of  the 
least  is  equal  to  |  of  the  greater. 

13.  The  sum  of  two  numbers  is  72.  If  the  larger  num- 
ber is  divided  by  the  smaller  number,  the  quotient  is  5. 
What  are  the  numbers  ? 

14.  The  sum  of  two  numbers  is  75,  and  when  the  smaller 
number  is  divided  into  the  larger  number,  the  quotient 
is  3  and  the  remainder  is  3.     Find  the  two  numbers. 

15.  The  largest  of  three  consecutive  odd  numbers  is 
divided  into  the  sum  of  the  other  two,  and  the  quotient  is 
1,  the  remainder  11.     Find  the  numbers. 

16.  A  boy's  age  is  one  third  of  that  of  his  father,  but  in 
4  years  the  boy's  age  will  be  two  fifths  of  the  father's  age. 
What  is  the  age  of  each  at  the  present  time  ?  ^ 

17.  A  certain  boy  is  one  and  one  third  times  as  old  as  his 
brother,  but  4  years  ago  he  was  twice  as  old.  Find  the 
present  age  of  each. 

18.  The  sum  of  the  ages  of  a  father  and  son  is  39  years, 
and  if  the  son  was  one  year  older  he  would  be  one  fourth 
as  old  as  his  father.     How  old  is  each  ? 

19.  A  can  do  a  piece  of  work  in  3  days  and  B  can  do 
the  same  work  in  4  days.  How  many  days  will  the  work 
require  if  both  work  together  ? 

(Hint  :  A  does  all  in  3  days,  therefore  he  does  |  of  it  in  1  day. 
Let  X  =  the  number  of  days  required  when  both  work  together. 
Then  how  much  of  the  work  will  both  together  do  in  1  day  ?) 


182  FRACTIONAL  EQUATIONS.      PBOBLEMS 

20.  A  can  do  a  piece  of  work  in  4  days,  B  the  same  work 
in  5  days,  and  C  the  same  work  in  6  days.  How  many 
days  will  it  take  to  perform  the  task  if  all  three  work 
together  ? 

21.  A  can  do  a  piece  of  work  in  7  days  and  B  can  do  the 
same  work  in  9  days.  C  is  called  in  to  help,  and  all  three 
together  do  the  work  in  3  days.  In  how  many  days  could 
C  alone  have  done  the  work  ? 

22.  A  certain  flock  of  sheep  contains  18  more  than  half 
the  number  of  sheep  in  a  second  flock.  In  both  flocks 
together  there  are  90  sheep.  Find  the  number  of  sheep 
in  each  flock. 

23.  A  man  spends  one  fourth  of  his  salary  for  household 
expenses,  one  seventh  for  rent,  and  one  tenth  for  miscel- 
laneous expenses.  How  much  is  his  annual  salary  if,  after 
the  above  expenditures,  he  has  left  1 1420  ? 

24.  A  man  left  two  thirds  of  his  estate  to  his  widow, 
one  twelfth  to  each  of  two  sons,  one  twenty  fourth  to  a 
brother,  and  iSOOO  to  his  church.  What  was  the  total 
amount  of  his  estate  ? 

25.  A  man  being  asked  his  age,  replied,  "  If  three  times 
my  age  be  decreased  by  14  years  and  the  result  be  divided 
by  my  age,  the  quotient  will  be  J."  What  was  the  man's 
age? 

26.  The  total  runs  made  in  a  baseball  game  was  11.  If 
the  winning  team  had  made  5  more  runs  and  the  losing 
team  2  more  runs,  the  quotient  of  the  winning  runs  divided 
by  the  losing  runs  would  have  been  2.  How  many  runs 
were  made  by  each  team? 


SIMPLE  FRACTIONAL  EQUATIONS  183 

27.  The  distance  around  a  rectangular  field  is  84  rods, 
and  the  length  of  the  field  is  |  the  width.  What  is  the 
length  of  each  side  of  the  field?  What  is  the  area  of  the 
field  ? 

28.  A  man  paid  13750  for  two  automobiles,  paying 
prices  for  each  such  that  |  the  cost  of  the  cheaper  one  was 
$  125  more  than  ^  the  cost  of  the  better  one.  What  was 
the  cost  of  each  ? 

29.  A  farmer  has  his  cattle  housed  in  three  barns.  In 
the  first  barn  there  are  3  more  head  than  \  the  whole 
number ;  in  the  second  barn,  2  less  than  |  the  total ;  and 
the  remainder,  19  head,  are  in  the  third  barn.  How  many 
head  are  there  in  all  and  how  many  in  each  of  the  first 
two  barns  ? 

30.  A  certain  fielder  played  in  seven  games  of  baseball 
and  made  three  more  hits  than  he  made  runs.  If  four 
times  the  number  of  hits  he  made  is  divided  by  the  number 
of  runs  increased  by  6,  the  quotient  is  3.  How  many  hits 
and  how  many  runs  did  he  make  in  the  seven  games  ? 

31.  An  estate  was  divided  among  three  heirs,  A,  B,  and 
C.  A  received  1500  more  than  one  fourth  of  it;  B,  1500 
less  than  one  half  of  it;  and  C,  $500  more  than  one  sixth 
of  it.  What  was  the  total  amount  of  the  estate  ?  What 
amount  did  each  receive  ? 

32.  Find  the  three  consecutive  even  numbers  such  that 
1  less  than  one  half  the  first,  plus  2  less  than  one  half  the 
second,  plus  3  less  than  one  half  the  third,  equals  15. 


CHAPTER   XV 

SIMULTANEOUS   SIMPLE   EQUATIONS. 
REVIEW 

If  X  and  y  represent  any  two  unknown  quantities,  then 

X  -\-  y  —  their  sum, 

X  —  y  =  their  difference. 

Suppose  we  assume  that  x=10  and  y=l.     Then 

a;  +  2/  =  17, 
x-y  =  ^. 

It  follows,  therefore,  that  w^e  have  two  equations  in 
which  the  same  unknown  quantity  has  the  same  value. 
Hence,  the  definition: 

112.  Simultaneous  Equations  are  equations  in  which  the 
same  unknown  quantity  has  the  same  value. 

113.  Simultaneous  equations  are  solved  by  obtaining 
from  the  given  equations  a  single  equation  with  but  one 
unknown  quantity.     This  process  is  called  Elimination. 

TRANSPOSITIONS  AND  DIVISIONS 

Before  taking  up  the  solution  of  simultaneous  equations, 
we  must  be  able  to  change  the  form  of  an  equation  in  two 

184 


TBAN8P0SITI0NS  AND  DIVISIONS  186 

unknowns  so  that  we  have  an  expression  for  one  of  the 
unknown  quantities  in  terms  of  the  other. 

(I)   Whei^  the  Coefficient  of  x  is  Unity 
Given  the  equation       x  +  2  y  =  7. 
Transposing,  a;  =  7  -  2  y. 

That  is,  7  —  2  2/  is  the  expression  for  the  value  of  x  in  terms  of  y. 
Again,  given  4  ?/  —  x  =  9. 

Transposing,  —x=9  —  4:y. 

Changing  all  signs,  a;  =  4  y  —  9. 

Oral  Exercise 

In  each  of  the  following  equations  give  the  value  of  x  in 
terras  of  ^ : 

1.  x-{-i/=b.  5.   i/-\-x  =  ll.  9.   —x—l7/=—3. 

2.  :r-f  By  =  10.  6.    — 4y  +  a?=13.  10.  x— Si/— 2  =  0, 

3.  x—2'i/=7.  7.   —2^  +  3^  =  4.  11.  y  — 4  — a;=0. 

4.  x  —  5^=S.  8.  2i/  —  x  =  —  l.  12.  ^  =  —  S~x. 

(II)  When  the  Coefficient  of  x  is  Any  Number 
Given  the  equation     3  x  +  2  y  =  5. 
Transposing,  Sx  =  6  —  2y. 

5    —   97/ 

Dividing  by  3,  x=  '- — -^^  •     Result. 


186     SIMULTANEOUS   SIMPLE  EQUATIONS.      REVIEW 
Oral  Exercise 

In  each  of  the  following  equations  give  the  value  of  x  in 
terms  of  y : 

1.  2a;+3?/  =  5.  7.  7a;-4?/  =  0. 

2.  ^x-y  =  ^.  8.   Zy-2x  =  ^. 

3.  2x  +  by  =  ^,  9.  2y-Zx-l  =  0. 

4.  3a;-22/=:-l.  10.   6a;-2i/  +  l  =  0. 

5.  5a:  +  3«/  =  -2.  11.    _2?/  +  6a;  +  l  =  0. 

6.  5a;-3  =  4y.  12.    -?/-32;-5  =  0. 

The  student  must  not  be  misled  by  the  fact  that  in  both 
of  the  foregoing  exercises  we  have  asked  for  the  value  of  x 
in  terms  of  y  rather  than  for  the  value  of  y  in  terms  of  x. 
The  value  of  either  in  terms  of  the  other  may  be  written 
without  difficulty.  The  x  term  was  selected  merely  to 
avoid  confusion.  In  the  following  exercises  the  student 
will  learn  to  select  whichever  x  ox  y  value  will  give  the 
probable  best  solution. 

ELIMINATION  BY  SUBSTITUTION 

Applying  the  principle  already  learned  in  the  foregoing 
oral  exercises,  we  will  now  illustrate  the  solution  of  a  pair 
of  simultaneous  equations  in  x  and  y, 

Ex.   1.     Solve  32:4-2?/  =  7;  2:r  +  «/  =  4. 

3^:4-2^  =  7.  (1) 

2  a;  +     3/  =  4.  (2) 

From  (2),  y  =  4  -  2  :r. 


ELIMINATION  BY  SUBSTITUTION  187 

(The  value  of  y  is  obtained  from  this  equation  because  its  selection 
avoids  a  fractional  result  in  transposing.) 
Substituting  in  (1), 

3  a;  +  2(4  -  2  a:)  =  7. 

3x+8-4a:  =  7. 

-  a:  =  -  1. 

a:=:l. 

For  the  value  of  y  we  substitute  in  one  of  the  original  equations. 
Hence,  substituting  in  (2), 

2  a:  4-  y  =  4. 

2(1)  +  y  =  4. 

y  =  2. 

X  =  l.) 

Hence,  [     Result. 

2/ =  2.) 

Ex.  2.     Solve  5a;+2?/  =  ll;3ic  +  4^  =  l. 

5x  +  2y  =  n.  (1) 

3a:  +  4y=    1.  (2) 

From  (2),  x  =  ^  ~^^. 

o 

Substituting  in  (1), 

From  which,  y  =  —  2. 


Substituting  in  (2),  3a:  +  4(-2)  =  l,         x  =  S. 
From  which,  a:  =  3.         y 


-2.> 


Result. 


From  these  illustrations  we  state  the  general  process  for 
elimination  by  substitution. 


188     SIMULTANEOUS  SIMPLE  EQUATIONS.      BEVIEW 

114.  From  one  of  the  given  equations  obtain  the  value  of 
one  unknown  quantity  in  terms  of  the  other  unknown 
quantity/. 

Substitute  this  value  in  other  equation  and  solve. 

Exercise  63 
Solve  and  verify  : 

1.  x-^i/  =  2,  10.  5  X- 7/  = -IS, 
Sx-\-2y  =  5.  dx-}-2^  =  0. 

2.  a:  +  «/  =  3,  11.  x  —  ^  =  —  7, 
2rr+3«/  =  8.  32:-2?/  =  -18. 

3.  a;  +  3/  =  5,  12.  5  ic  +  ^  =  15, 
3a:  +  2y=12.  22:-3^=6. 

4.  5a;+T^=12,  13.  5^ -a:  =  4, 
3a:  +  y  =  4.  2a;-^  =  -8. 

5.  4a;+2?/  =  22,  14.  4a;+3y=9, 
5 x  —  2^  =  5,  X—  6  i/  =  0. 

6.  4:i/-Sx=16,  15.  6x-2  7/=7, 
Sx+2i/  =  26.  3a:  +  4y  =  12. 

7.  4a;-9«/  =  -5,  16.  6  re -3?/ =2, 
5t/-2x=5.  2x-Sy  =  Q. 

8.  6  a;  +  5  2/  =  4,  17.    -  5  a;  +  3  ?/  =  2, 
2a;-3?/  =  -8.  6^  +  5a:=l. 

9.  Sx  +  2  =  -l y,  18.  8a^+3?/  +  5  =  0, 
2a;=3^-9.  2a;=3y. 


ELIMINATION  BY  COMPABISON  189 

ELIMINATION  BY  COMPARISON 

The  process  of  elimination  by  comparison  is  illustrated 
in  the  following : 

Ex.1.     Solve  5a;+2y  =  9;  2x-{-Si/  =  S. 

From  each  equation  obtain  the  value  of  one  unknown  in  terms  of 
the  other,  selecting  the  same  unknown  for  each  value. 

5  a;  4-  2  y  =  9.  (1) 

22:+3y  =  8.  (2) 

From  (1),  x=     ~     ^. 

5 

From  (2),  x  =  ^~^^' 

9-2w8-3w 


Hence,  Art.  58, 

O  J!i 

Clearing,  2(9  -  2  t/)  =  5(8  -  3  y; 

18-4y  =  40-15  ?/. 

y  =  2. 

Substituting  in  (1), 

5x4-2(2)  =  9. 

x  =  l. 

x=  1 

y 


:;;) 


Result. 


From  this  illustration  we  state  the  general  process  for 
elimination  by  comparison. 

115.   From  each  equation  obtain  the  value  of  the  same  un- 
known quantity/  in  terms  of  the  other  unknown  quantity. 
Place  these  values  equal  to  each  other  and  solve » 


190     SIMULTANEOUS  SIMPLE  EQUATIONS.      REVIEW 

Exercise  64 

Solve  and  verify  : 

1.  2x+Sy  =  lS,  9.  6a;  +  y=ll, 
2x-{-^=7.  ^-6x  =  -lS, 

2.  3  2^4-2^  =  25,  10.  5 re +  2 3/ =  20, 
2x-y  =  5.  Sx-4:i/  =  12, 

3.  3rr  +  5i/  =  29,  11.  4a:+3y  =  ll, 
3rr+2?/=17.  2x  +  5^  =  16. 

4.  4:r-y  =  l,  12.  2a;+3^=9, 
3a;  +  2«/=20.  4a;+92/  =  19. 

5.  3a;-2i/  =  13,  13.  6  a; -2?/ =3, 
4a;+3y=6.  3r?;  +  2^=l. 

6.  5a;4-3y  =  —  5,  14.  3a;  —  4«/=  —  9, 
7a:H-2?/  =  4.  3?/  +  3a;=-7. 

7.  22;  +  3?/=^18,  15.  8a;4-3y=-4, 
3a;-2y=l.  4a;-5?/=-2. 

8.  3rr  +  y=l,  16.  2  ?/ +  5 a;  =  2, 
Ua:+3y  =  l.  10a;  +  32/-l  =  0. 


ELIMINATION  BY  ADDITION   OR   SUBTB ACTION     191 


ELIMINATION  BY  ADDITION  OR  SUBTRACTION 

This  method  of  elimination  produces  a  solution  free  of 
fractions,  and  is  frequently  useful  when  the  given  coeffi- 
cients are  small  numbers. 

(1) 

(2) 


Ex.  1.    Solve 

3a;+    2^  =  11 
4a;-    3i/=    9 

Multiply  (1)  by  3, 
Multiply  (2)  by  2, 
Adding, 

9a; 4-    62/  =  33 

8a;-    6^  =  18 

17a;             =51 

Substituting  in  (1), 


a;  = 


::) 


Result. 


Ex.  2.   Solve 

Multiply  (1)  by  5, 
Multiply  (2)  by  2, 
Subtracting, 

Substituting  in  (1), 


3a;  + 
2a;  + 


2^  =  10 
5«/  =  14 


15a;  +  102/  =  50 
4a;  +  10^  =  28 


11a; 


22 
x=    2. 

y 


ii] 


(1) 

(2) 


Result. 


Hence,  for  eliminating  by  addition  or  subtraction : 

116.  Multiply  one  or  both  given  equations  by  the  smallest 
numbers  that  will  make  the  coefficients  of  one  unknown  quan- 
tity equal. 

If  the  signs  of  the  coefficients  are  unlilce^  add  the  equa- 
tions ;  if  like.,  subtract  the  equations. 

For  practice  the  student  should  solve  the  equations  in  Exercise  63 
by  the  method  of  addition  and  subtraction. 


192    SIMULTANEOUS  SIMPLE  EQUATIONS.      BEVIEW 

FRACTIONAL  SIMULTANEOUS  EQUATIONS 

Clear  the  following  simultaneous  equations  of  fractions, 
and  solve  the  resulting  equations  by  the  method  of  substi- 
tution or  by  the  method  of  comparison  : 


Exercise  65 

1. 

2      3 

7. 

2.+1      y+1^ 
3       '      5 
x+1     y+l_      . 
5            2 

2. 

2        5 

8. 

3-+1     5  +  2^  =  2, 

2*  ^Y^  4='>- 

3. 

2      3       ' 

4^  2 

9. 

y-J  +  1   -+J-0. 

b                         d 

4. 

2        4 

10. 

3v-5a;     3 
2        ==8' 
2a;  +  4.y_4 
a;-32/       3 

5. 

2            3 

=  8, 

11. 

rr  +  ^  +  l      2 
12           3' 

^  +  1  +  2^= 
4      '      5 

=  2. 

8            2 

6. 

x  +  2     ,y-'4 
4      '      3     ~ 

=  3, 

12. 

22:H-?/4-l_3 
3rr-?/  +  2      6' 

a:-3     .y-2_ 
3      '      5    " 

=  2. 

4a:-3«/  +  2_5 
2a;-3^  +  l      3 

SIMULTANEOUS  EQUATIONS  193 

PROBLEMS  PRODUCING  SIMULTANEOUS  EQUATIONS 

Problems  involving  two  unknown  quantities  are  readily 
solved  by  the  use  of  two  different  letters,  x  and  ?/;  and 
the  pair  of  simultaneous  equations  obtained  from  the  con- 
ditions of  the  problem  is  treated  in  accordance  with  either 
of  the  methods  already  learned.  The  following  problems 
require  the  use  of  two  unknown  quantities  and,  in  each 
problem,  two  equations  are  necessary  for  a  solution. 

Exercise  66 

1.  The  sum  of  two  numbers  is  15,  and  their  difference 
is  3.     What  are  the  numbers  ? 

Let  X  =  the  greater  number, 

y  =  the  less  number. 
Then  x-\-y  =  15.  (1) 

x-y  =  3.  *  (2) 

From  (1),  y=15-x. 

Substituting  in  (2),  x  —  (15  —  x)  =  S. 
x-15  +  x  =  3. 
2a:=18. 
x  =  9. 
Substituting  in  (1),  9  +  y  =  15. 

y  =  Q. 
Hence,  x  =  9,  the  greater  number. " 

y  =  Q,  the  less  number. 


Result. 


2.   The  sum  of  two  numbers  is  29,  and  their  difference 
is  5.     What  are  the  numbers  ? 

F.  H.  S.  FIRST  TEAR  ALG.  —  13 


194     SIMULTANEOUS   SIMPLE  EQUATIONS.      REVIEW 

3.  The  difference  between  two  numbers  is  15,  and  the 
quotient  of  the  larger  divided  by  the  smaller  is  6.  Find 
the  numbers. 

4.  The  difference  between  two  numbers  is  40,  and  if 
the  larger  number  is  divided  by  the  smaller,  the  quotient 
and  the  remainder  are  each  4.     Find  the  two  numbers. 

5.  One  half  the  sum  of  two  numbers  is  14,  and  4 
times  their  difference  is  16.     Find  the  numbers. 

6.  One  half  the  sum  of  two  numbers  is  2  more  than 
their  difference.  Five  times  the  smaller  number  equals 
twice  the  larger  number.     Find  the  numbers. 

7.  There  are  two  numbers  such  that  3  times  the  first 
added  to  4  times  the  second  equals  37,  and  4  times  the  first 
less  3  times  the  second  equals  —  9.     Find  the  numbers. 

8.  If  3  is  added  to  the  greater  of  two  numbers  and  2  is 
subtracted  from  the  smaller,  the  sum  of  the  results  will  be 
14.  If  the  greater  number  is  divided  by  2,  the  quotient  is 
1  less  than  the  smaller  number.     Find  the  numbers. 

9.  Add  the  least  of  two  numbers  to  one  half  the 
greater  and  the  sum  is  12.  Subtract  the  least  number 
from  one  fifth  the  greater  and  the  difference  is  —  5. 
What  are  the  numbers? 

10.  If  the  greater  of  two  numbers  is  divided  by  the 
less,  the  quotient  is  3  and  the  remainder  6.  If  4  times 
the  less  number  is  divided  by  the  greater,  the  quotient  is 
1  and  the  remainder  6.     Find  the  numbers. 

11.  The  sum  of  two  numbers  is  divided  by  their  differ- 
ence, and  the  quotient  is  5.  If  the  sum  of  the  numbers  is 
divided  by  12,  the  quotient  is  f .     What  are  the  numbers  ? 


SIMULTANEOUS  EQUATIONS  195 

12.  If  1  is  added  to  both  the  numerator  and  denomi- 
nator of  a  certain  fraction,  the  resulting  fraction  is  |.  If  1 
is  subtracted  from  both  the  numerator  and  denominator  of 
the  fraction,  the  resulting  fraction  is  J.     Find  the  fraction. 

(Hint  :    Let  -  =  the  given  fraction.     Then  ^  "^     =  -,  etc.) 
y  2/4-  1      4 

13.  A  certain  fraction  becomes  ^  when  5  is  added  to 
the  denominator,  but  if  5  is  added  to  the  numerator,  its 
value  is  |.     What  is  the  fraction  ? 

14.  If  1  is  added  to  both  numerator  and  denominator 
of  a  certain  fraction,  the  resulting  fraction  is  ^.  If  the 
original  fraction  is  inverted  and  3  is  added  to  both  numer- 
ator and  denominator,  the  resulting  fraction  is  ^,  What 
is  the  given  fraction  ? 

15.  The  numerator  of  a  certain  fraction  is  multiplied 
by  3  and  the  denominator  by  2,  the  resulting  fraction 
being  f ;  but  when  the  numerator  is  increased  by  2  and 
the  denominator  increased  by  2,  the  resulting  fraction  is  ^. 
What  is  the  fraction  ? 

16.  When  the  numerator  of  a  certain  fraction  is  doubled 
and  the  denominator  increased  by  3,  the  result  is  J.  If 
the  denominator  is  doubled  and  the  numerator  decreased 
by  1,  the  result  is  -^q.     What  is  the  fraction  ? 

17.  A  father  is  32  years  older  than  his  son,  but  8  years 
ago  the  father  was  5  times  as  old.  What  are  the  present 
ages  of  each  ? 

(Hint  :  Let  x  =  the  father's  age  now,  y  =  the  son's  age  now.  Then 
X  —  y  =  32.     And  x  —  S  =  5(y  —  8).     State  and  solve.) 

18.  Three  years  ago  a  certain  boy  was  twice  as  old  as 
his  sister,  but  at  the  present  time  the  sum  of  their  ages  is 
30  years.     Find  the  ages  of  each  at  the  present  time. 


196     SIMULTANEOUS  SIMPLE  EQUATIONS.      REVIEW 

19.  The  sum  of  the  ages  of  John  and  William  is  42 
years,  but  one  third  of  John's  age  is  1  year  less  than 
one  half  of  William's  age.     What  is  the  age  of  each  ? 

20.  Four  years  ago  A  was  twice  as  old  as  B,  but  in 
6  years  A's  age  will  be  nine  sevenths  that  of  B.  What  is 
the  age  of  each  now  ? 

21.  A  lady  bought  10  yards  of  cloth  and  4  yards  of 
silk,  paying  f  32  for  the  whole.  Later,  at  the  same  prices, 
she  bought  5  yards  of  cloth  and  6  yards  of  silk  for  128. 
What  was  the  cost  of  the  cloth  and  of  the  silk  per  yard  ? 

(Hint  :  x  =  cost  per  yard  of  the  cloth  in  dollars,  y  —  cost  per  yard 
of  the  silk  in  dollars.     Then  10x  +  4:y=  .32,  etc.) 

22.  A  man  bought  4  horses  a^nd  3  wagons  for  #780,  and 
later,  5  more  horses  and  2  more  wagons,  paying  $870  for 
the  second  lot.  The  cost  of  both  horses  and  wagons  was 
the  same,  respectively,  in  each  lot.  What  was  paid  for 
each  ?  [ 

23.  Five  pounds  of  coffee  and  3  pounds  of  rice  cost 
f  1.80,  and  4  pounds  of  coffee  and  7  pounds  of  rice  cost,  at 
the  same  rates,  $1.90.  W^hat  is  the  cost  of  the  coffee  and 
of  the  rice  per  pound  ? 

24.  A  boy  has  32  marbles  in  two  pockets.  One  half  of 
the  number  in  one  pocket  is  2  less  than  the  number  in  the 
other  pocket.     How  many  marbles  has  he  in  each  ? 

25.  A  said  to  B,  "  If  my  weight  divided  by  3  is  added 
to  your  weight  divided  by  4,  the  total  is  90  pounds." 
B  replied,  "  If  one  eighth  of  my  weight  is  diminished  by 
one  sixth  of  your  weight,  the  result  is  5  pounds."  What 
was  the  weight  of  each  ? 


GENERAL   REVIEW  197 

GENERAL  REVIEW  — GROUP  V 

FRACTIONAL  EQUATIONS 

Clearing  of  Fractions 

Multiplying  both  members  of  a  fractional  equation  by  the 
lowest  common  multiple  of  their  denominators  removes  the 
fractions  from  the  equation. 

Signs 

The  sign  of  a  fraction  becomes  the  sign  of  its  numerator 
when  the  equation  is  cleared  of  fractions. 

Simultaneous  Simple  Equations 

For  each  solution  we  must  have  as  many  equations  as  there 
are  unknown  quantities. 

elimination 

We  make  use  of  two  methods  : 

1.  By  Substitution.  In  which  we  replace  an  unknown 
quantity  in  one  equation  by  the  value  of  that  unknown  ob- 
tained from  the  other  equation. 

2.  By  Comparison.  In  which  we  form  an  equation  with 
the  values  of  one  of  the  unknowns  in  terms  of  the  other^  these 
values  being  obtained  from  the  given  equations. 

FINAL  REVIEW 
Exercise  67 
A 

1.  What  is  the  sum  of  (a-\-b')x  —  {a—b^y-^2(a-^c)z 
-  3  (a  +  5)  :c  +  4  (a  -  6)  y  -f  3  (a  +  0  25  +  3  (a  +  ^>)  a;  - 


198     SIMULTANEOUS   SIMPLE  EQUATIONS.      REVIEW 

2.  Take  the  sum  of  2a^-a^  +  a-2>  and  2a^^Za^  + 
^  a  +  Q  from  the  sum  of  3  a  —  4  a^  -f  -i  a^  +  2  and  1  +  3  a^ 
-  3  a3  4-  4  a. 

3.  Take  the  result  of  [(^2- ^4- 1)- (2a- ^2  + 2)] 
from  the  sum  of  a  +  2  —  a2  and  a?—Za—l. 

4.  Take  a;2_^  2  rcH- 3 'from  2^ +  3  re +2  and  multiply 
the  result  by  the  sum  of  4  a:  —  10  —  a;2  and  11  -\-  x^  —  ^  x. 

5.  Divide  the  sum  of  a;^  —  7  a;2  +  14  2;  —  17  and  2  a^  — 
32:  +  2  by  (2;  -  3)2 +  4  (a: -1). 

6.  Subtract  Q  —  x  —  ^x^  from  the  product  of  (—  3  + 
2:2-2:)  by  (x-2  +  x^). 

7.  Divide  4a*-9a2+6a-l  by  2a2_3a  +  l  and 
multiply  the  quotient  by  2  a  —  3. 

8.  Simplify  [(3  ««  +  a^  -  2  a)  +  (a^  _  a*  +  1)  -  (2  a^  + 
2  ^5  _  a*  —  2  a  +  2)]  and  divide  the  result  by  (a2  +  a  +  1) 
(a2_a  +  l). 

B 

Simplify : 

9.  a:-[l  +  |2:-(l  +  2:)S]. 


10.  _J_[_(^_^_1)]|. 

11.  a  —  [  —  (  —  ;  —  a  J  —  a)  —  4  a] . 

12.  c-(2d-(i-c-\l-2d-ic-T^^)\')-'V). 

13.  -[c+ Jc-(c-l)-(c  +  l)-ci -(?]. 

14.  22:-[2:  +  l-S2:  +  l-a-(2;+l4-a-w)|]. 

15.  (a  +  l)2-(2a-l)2+3a(a-2). 

16.  a(a-l)2- a(a2  +  a-l)-4a(l-a). 

17.  (a  +  l)(2;+l)-  (a-l)(2:-l)-2(a  +  a;-l). 


GENERAL  REVIEW  199 

18.  2(a+l)(a-5)  +  3(a-2)(a  +  5)-(4a+5)(a-8). 

19.  (a-l)3-(a+l)3  +  7a2  4-2. 

20.  Qi-\-x+iy-a(x-\-l-^a)-x(a-\-x-V), 

C 

Solve  and  verify : 

21.  5a;-3(2a:-ll)-(rr-l)  =  0. 

22.  4a;+ [2  a;- (a:H-l)]  =a;-5. 

23.  2a;2-(2;  +  l)  =  22;(a;-3). 

24.  3  n;  +  (:z^  -  1)  (2  2^  +  1)  =  (2  a;  -  1)  (a;  +  1)  -  1. 

25.  (2a;4-l)2-3a;(ic+2)-(a;-5)2  =  0. 

State  and  solve : 

26.  One  number  is  4  times  another  number,  but  if  the 
smaller  number  is  increased  by  17  and  the  larger  num- 
ber decreased  by  22,  the  results  are  equal.  Find  the 
numbers. 

27.  A  is  26  years  older  than  B,  and  A's  age  is  as  many 
years  more  than  34  as  B's  age  is  years  less  than  34.  Find 
the  age  of  each. 

28.  A  man  has  four  sons,  each  3  years  older  than  the 
next  youngest.  The  age  of  the  oldest  is  21  years  less 
than  the  sum  of  the  ages  of  all  four.  Find  the  age  of 
each. 

29.  A  man  paid  a  bill  of  '$7.20,  using  dimes,  quarters, 
and  dollars.  He  used  twice  as  many  quarters  as  dimes 
and  three  times  as  many  dollars  as  dimes.  How  many 
coins  of  each  kind  did  he  use  ? 


200     SIMULTANEOUS  SIMPLE  EQUATIONS.      BEVIEW 

30.  Two  automobilists  start  toward  each  other  from 
towns  88  miles  apart.  The  first  travels  at  a  rate  of  10 
miles  an  hour  and  the  second  at  a  rate  of  12  miles  an  hour. 
In  how  many  hours  will  they  meet  ? 

31.  A   man  rows  downstream  at  a  rate  of  8  miles  an 
our,  but  upstream  his  rate  is  but  4  miles  an  hour.     How 

.ar  downstream  can  he  go  and  return  in  6  hours  ? 

32.  A  boy  agreed  to  work  50  days  at  10.50  a  day  for 
each  day  he  worked.  For  each  day  he  was  idle  he  agreed 
to  forfeit  fO.25.  He  received  $22  when  the  settlement 
was  made.     How  many  days  did  he  work  ? 

D    '     • 

Find  the  value  of : 

33.  (a-l)2-a(a  +  l) +^(a-l)  when  a  =2. 

34.  (x-Sy-(x-\-S)(x-S)-(x-\-Sy  whena;=-2. 

35.  2  a  (^a  -{-  X  —  1}  —  (^a  —  x){2  a  -\-  x^  when  a  =  3  and  x 
=  -3. 

36.  m  (m  +  H  +  1)  —  (w  —  n  +  1)  (m  +  w  —  1)  when  m  = 

0  and  n  =  -- 

2 

37.  3a;(a:+l)2-3(a;+l)3+3(a;+2)(a;-5)   when   x 

2 

—  ~  3- 

Substitute  in  the  following  formulas  and  obtain  required 
values : 

38.  If  a  =  4,  w  =  6,  c?  =  2,  find  the  value  of  I  in 


GENERAL  REVIEW  201 

39.  If  a  =  -2,  Z=-128,  r  =  - 2,  find  the  value  of  aS' in 

a     rl—  a 

40.  If  a  =  3,  5  =  4,  <?  =  —  7,  find  the  value  of  x  in 

"""  2^ 

41.  If  a  =  6,  6  =  —  1,  c  =  —  35,  find  the  value  of  x  in 


_5_V62-4ac 
a;  = 

2a 

42.  If  a  =  1,  <f  =  2,  w  =  11,  find  the  value  of  aS'  in 

43.  If  (?  =  3,  ?  =  40,  n  =  13,  find  the  value  of  a  in 

1  =  a-\-(n  —  V)d. 

44.  Given  a  —  —  b^  Z  =  15,  /S'=  105,  what  is  the  value  of 
winAS'=^(a-|-Z)? 

A 

45.  How  far  will  a  body  fall  in  6  seconds  if  g  in  the 
formula  aS'=-|  ^^2  is  32  ft.? 

46.  How  many  turns  are  made  by  a  wheel  4  feet  in 
diameter  in  going  3  miles  ? 

47.  What  is  the  area  of  a  circular  pool  whose  diameter 
is  100  feet  ? 


202    SIMULTANEOUS  SIMPLE  EQUATIONS,      BEVIEW 

E 

Multiply  by  inspection: 

48.  (2a  +  3a:)2.  54.  (4a:- 1) (3 a;- 5). 

49.  (4a-7«/)2.*  55.  (2x^-4:')(Sx^-\-l}. 

50.  (3«  +  5a;)(3a-5a;).  56.  (5a:- 8) (4  a;  +  3). 

51.  (a;+T)(a:-10).  57.  (2 a:  +  7) (3 a:- 10). 

52.  (a^-4)(a^H-13).  58.  (a +  2  a; -3)2. 

53.  (2a:  +  3)(3a:  +  4).  59.  (2a3  +  3  ^2  + ^  +  2)2. 

Divide  by  inspection : 

60.  4  a2  _  25  by  2  a  -  5.  66.  27  a^  _  1  by  3  «  -  1. 

61.  a^  -  64  by  a  -  4.  67.  a*  -  1  by  «  -  1. 

62.  ^3  _|.  125  by  a  +  5.  68.  a*-lbya  +  l. 

63.  8  a^-  27  by  2  a  -  3.  69.  a*  -  a:^  by  «  -  a:. 

64.  125  4-  a:^  by  5  +  a;.  70.  a^  —  a:^  by  a  —  a:. 

65.  64  -  a3  by  4  —  a.  71.  a:^  _|_  32  by  a;  +  2. 

F 

Factor : 

72.  a^-ia.  76.  a^-64a;. 

73.  27  63(^3  +  1.  77.  12m'-m  +  x^-m\ 

74.  16a2-40a+25.  '  78.  ar^- lla:2__  423,. 

75.  ^2  _  19  ^_  120.  79.  a;2-a2_i0a_25. 


GENEBAL  REVIEW  203 

80.  ^x^  +  Qx-^b.  88.  m3-8-3(w-2). 

Q1.  a^-x^  +  x-l,  89.  4  2^ +  17  a: +  4. 

82.  4(2; -1)2 -9.  90.  ax-\-a-\-^x  +  ^. 

83.  39-10c-c2.  91.  216  m3- 125. 

84.  4:c*-32a;.  92.  x-lbx^  +  ZQa^. 

85.  7c2_98c  +  343.  93.  25  2:2_4(^_2)2. 

86.  b-10x-bc^  +  bx\  94.  10a:2-63  +  17a;. 

87.  6r2-a;-2.  95.  a:5_a;3_^2^1. 

G 

Find  the  H.  C.  F.  and  the  L.  C.  M.  of: 

96.  9(c  +  (^)2,  30((?*-(?*)^  45(c2-2c^  +  (i2^. 

97.  x{x-iy,  0:2(2: +1)2,  x^x^-l). 

98.  3(:r  +  l)(2:+2),  4(:z:  +  l),  6(a:+2). 

99.  m2-m-20,  m2+8m  +  16,  m2  +  10m+24. 

100.  9(^:2-:?: -2),   6(^-2a:'-3),   12(^-5  2: +  6). 

101.  3(^3  +  53),  6(a2-52),  9(^3-53). 

102.  22:2-a:-l,  22^-3a;-2,  ?>x^-^x+Q. 

103.  xy  ■\- X -\- y  +  1^  xy  —  X'\-  y  —\,  xz  -\-  z  -\-  x  -\- 1. 

104.  (^2-52)2,  7(<z  +  5)3,  6(a*-54). 

105.  a^-(x-{-iy,  (a-\-xy-h  (^  +  1)2-2:2. 


204    SIMULTANEOUS  SIMPLE  EQUATIONS.      REVIEW 

H 

Reduce  : 

"*'•   Zi — 5 — I ^ o 5  to  lowest  terms. 

^4  .  Q    Q  ,  o    9  to  lowest  terms. 

"^*   ~T"^ — 2r~; — m  .   iQ  to  lowest  terms. 

^  _  ^2  _  20 
^^^'   "TT — TT"  ^^  ^  mixed  expression. 

~2 T"  *^  ^  mixed  expression. 

111.  rr^  —  07  4- 1 to  an  improper  fraction. 

37+1  i-       x- 

112.  a — - —  a  +  2  to  an  improper  fraction. 

113.  c-\-  d—1 — -—  to  an  improper  fraction, 

c+d+1  ^    ^ 


I 

Collect : 

114.  1         •  -1 


115. 


x—2a     x^—5ax+6a^     x—Sa 

g  +  4      g-4      15g+2 
e-4:      c  +  4.     c^-ie' 


116.    -1-^+  1 


a2+3a  +  2     a2  +  5a  +  6 
117.        ^<^       I        «  2  a 


2a  +  2      3a-3      6a2_6 


118. 


119. 


120. 


121. 


122. 


GENERAL   REVIEW 

1 

1         , 

1 

(a+1) 

{a-\-iy     ( 

:«+i)3 

x^-1 

2x2       x-1 

x-1      : 

r-l      x+1 

2 

2 

4 

a?-l 

l  +  a2      (^2. 

-IX^'^  +  I) 

7x 

a:-2 

1 

^-8  ' 

a:2+2a;  +  4 

x-2 

a^ 

,       1 

x 

{x-{-iy 

'  37+1       (:r 

+  1)^ 

2x 

3 

2 

205 


123 

'    3(a;-2)2      2^  +  4     a:2-4 


Simplify : 

124.   [a  + 6' —J. 

io«     a2-4a  +  3    a^-2a 


a^  —  6  a  +  5     a^  —  a^ 

127.  r^«^Vfi-i+i\ 

\am  —  ax  ■{-  mxj    \a     m     xj 
128.    fl-lY-^V^^^^'. 

U^     Ai  +  W     i-:r2 


3  2: +3 


129.  fx-i  +  _^y/|^    1' 

I  ^r+iy    V3      2      6, 


206     SIMULTANEOUS   SIMPLE  EQUATIONS.      UEVIEW 
2^2-3^^  +  2  2:2-16 


130. 


131. 


2;2-5a;4-6  *  a^  +  ^-12' 
c  +  d        c^-d^    .  c2+^ 


132     jx-iy-a?'    x^-^a-iy 


133. 


^2—  9 _^ r 6t9^  —  3 n      aa; 4-  ^  +  3  a;  +  31 
c2  —  4  *  [_cm  +  2  m      c?a;  +  ^  —  2  a;  —  2  J* 


Solve  and  verify  : 

x-1     x-{-2     x-2     x  +  1 


134. 


6 


135.  2(£+3)^3_2^+l^ 

5  2 

136.  i^-|(.-l)-£±l  =  0. 


137. 


:r  +  l      2^-1  8 


X  --l      x+1      x^—1 


138.    ^^-±1-=    ^-^    + 


2a  +  6     3a-9     6a2_54 

State  and  solve: 

139.  The  difference  between  the  numerator  and  denomi- 
nator of  a  certain  proper  fraction  is  5.  If  4  is  added  to 
both  the  numerator  and  the  denominator,  the  resulting 
fraction  is  -^2  •     What  is  the  original  fraction  ? 


GENERAL  REVIEW  207 

140.  A  certain  number  is  decreased  by  12  and  the 
remainder  divided  by  4.  If  the  resulting  quotient  is 
increased  by  7,  the  sum  is  the  same  as  if  the  original 
number  had  been  increased  by  7  and  then  divided  by  3. 
What  is  the  original  number  ? 

141.  A  boy  can  mow  a  lawn  in  7  hours  if  he  works  alone, 
and  a  second  boy  can  do  the  same  work  alone  in  6  hours. 
In  how  many  hours  can  the  boys  together  accomplish  the 
task  ? 

142.  A  gentleman  orders  a  bill  of  books.  Three  less  than 
half  the  whole  number  are  fiction,  2  more  than  a  third  of 
the  whole  are  history,  and  3  less  than  a  quarter  of  the 
whole  are  biography.  How  many  books  are  there  of  each 
kind? 

143.  Divide  $  630  among  A,  B,  and  C,  so  that  A  shall 
receive  half  as  much  as  B,  and  C  two  thirds  as  much  as  A 
and  B  together  receive. 


Solve  by  substitution  :  Solve  by  comparison  : 

144.  2x-\-i/  =  3,  147.  4:x  +  ^i/  =  —  29, 
Sx-2i/  =  4:,  5ic-4y=-13. 

145.  4a;-5^  =  2,  148.   ll:r+7?/4-29  =  0, 
2  :r  +  3  ^  =  12.  3  a;  -  33  =  -  11  y. 

146.  22;  +  5?/-4  =  0,  149.    -10  +  5x=S7/, 
Sx-\-2i/-{-5  =  0.  1 2/-4:  =  -2x. 


208     SIMULTANEOUS  SIMPLE  EQUATIONS.      REVIEW 
Solve  by  either  method  : 

153.  ^-y+^=2, 

4 

154.    ^-1=^, 
5  2 

3     "^  4     ' 


150. 

I* 

1-^ 

X 

10" 

-!=->■ 

151. 

3 

-¥=-*• 

3a: 
2 

-I-"- 

152. 

a:  — 

2 

i.^=., 

155. 


^  +  .y  +  3^2 

x-y-2      3' 

a;+l  .  .y  +  1^3  ^_-^jf2^4 

2  5  '  a^  +  y-3     3* 


State  and  solve  : 

156.  Find  two  numbers  whose  sum  is  37. and  whose 
difference  is  13. 

157.  One  fourth  of  a  number  added  to  one  third  of  a 
second  number  gives  a  sum  of  8.  One  third  of  the  first 
number  increased  by  one  fifth  of  the  second  gives  a  total 
of  7.     What  are  the  numbers  ? 

158.  If  the  sum  of  two  numbers  is  divided  by  7,  the 
remainder  is  4  and  the  quotient,  6  ;  but  if  the  difference 
between  the  numbers  is  divided  by  7,  the  remainder  is  2 
and  the  quotient,  4.     Find  the  numbers. 


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